Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications

dt

= -P • (AxQp - (lkAo) + - T r (M[fifc, AJ) In particular,

23

dP — • A0 = P • nkA0 at

= -ilkP

-

• A0 ,

and therefore, —- = -flkP. Equivalently, —- = [fifc,P]. at at It then follows that Ti(M[Slk,Ai]) = Ti([M,Qk]Ai) from which we get that

To obtain the equation for M(t) we need to recognize that Aa • b can be written as - -Trace ((a A b)A) for any vectors a and b, were a A b denotes the antisymmetric matrix defined by (a A b)x = (a • x)b — (b • x)a. Therefore,

-^Tr ( ^ f V ) =-\Tr((P*flp)A1) + lTt{[M,nl]A1) from which it follows that ^

= [nt,M] + P A O p ,

a n d ^ = [tt fc ,P]

(3.5)

are the Hamiltonian equations on the semi-direct product. Let us now look at the conservation laws that are due to the left-invariant symmetries. As already noted in 5 , the solutions of equations (3.1) evolve on co-adjoint orbits of G i.e., E(t) = Ad*(t)(^(0)). Therefore, any right invariant vector field on G is a symmetry for the Hamiltonian H. Indeed, let X(g) = Ag be any right-invariant field. Then its Hamiltonian Hx is given by Hx(t,g) = tig^Ag) = A d ; _ , ( 0 ( 4 ) , and hence Hx{l(t),g(t)) = Ad^Ad^O))^) = £(0)(A). Consequently, Hx is an integral of motion for H. For semi-simple Lie groups, the condition that l(g~1Ag) is constant for any A in g is expressed equivalently as g^Litfg-1

(t) = constant

(3.6)

for any integral curve of (3.1). In particular, (3.6) implies that the spectrum of L(t) is constant along the solutions.

24

The spectral invariants of L(t) are also known as Casimir functions on g. Evidently, they are conserved along the flow of any left-invariant Hamiltonian H. It is easy to see that Trace Lk, k > 1 is a Casimir function on any semisimple Lie group, because ^-Trace(L fc ) = k (Lk-\ at \

^ \ = k{Lk-\[dH,L}) at J

= k(dH, [Lk~\L\)

=0.

Of course, the trace of Lk is related to the coefficients in the characteristic polynomial of L(t). For semi-direct products the adjoint orbits are not in the same correspondence with co-adjoint orbits and hence condition (3.6) takes a slightly different form. We recall equation (3.5)

£ = m..*]+»Afl,. § together with — = gdH = g($lp + $lk), where 1 0

9=

[xRl-

Hence, dR dx ^ - 3 - = Rilk and — = Rilp dt dt It follows that

^-R{i)P(t) = -RtlkP + RQkP = 0 dt

and jiRMR-1

+xARP)=

R(nkM

- Af fifc)iT1 + R([ilk, M] + P A(lp)

+ Rtip A RP = 0 . Thus, R{t)P(t) -

constant, and R{t)M{t)R~l

(t) + x(t) A RP = constant

(3.7)

25

are the conservation laws on the semi-direct product due to left-invariance. Let us now exploit the conserved quantities. 3.3

The Conservation Laws

As already remarked the quantities Tr(L*) are conserved along the trajectories of a left-invariant Hamiltonian system on a semi-simple Lie algebra g. Remarkably, some of these conserved quantities also persist on semi-direct products as we shall presently see. Let us begin with non-Euclidean cases. The extremal curves L(t) are written as sums M + P with pn\

/ 0 -pj epi

P(t) =

0 \£Pn

and M(t) an n x n antisymmetric matrix. It will be convenient to write

£(*) = with P = (pi,...

,pn)T-

o -PT eP

M

Then,

||L|| 2 = (L,L) = - - T r L 2 = e | | P | | 2 + (M,M)

.

Let h = e||£|| 2 = ||P|| 2 + e(M,M). It follows from (3.7) that ||P(t)|| 2 is constant along the extremal curves on the semi-direct products. Hence, we may simply let e = 0 in the above expression to get the appropriate integral of motion on the semi-direct product. Let us now calculate the trace of L4. Then, 2 _ f-e\\P\\2 eMP

M

2

-PTM ~eP®P

with a b denoting the matrix (a ® b)x = (b • x)a. It follows that

26

TrL 4 = 2||P|| 4 + TraceM 4 + 4£||MP|| 2 But TraceM 4 = 2(M,M) 2 , and I\ = ||P|| 4 + <M, M>2 + 2e(M, M)\\P\\2 Hence £

2

I

r2 +, -I T TV i-K/t » M I I D I I 2 + , 11^6112 -/f r Lr 44 \) = -{M,M)\\Pf \\MP\\

It follows that I3 = -(M,M)\\P\\2

+ \\MP\\2

is an integral of motion independent of e. It is now an easy matter to verify that I3 is an integral of motion on the semi-direct product as well. Let us single out two cases that will be needed later on. In the three dimensional case, 0 - M 3 M2 M = I M3 0 -Mi 0 k - M 2 Mi 2

2

2

fpi V2 \P3J

2

2

and \\MP\\ = \\M x P\\ = ||M|| ||P|| - (M • P) . Therefore, h = - ( M i p i + M2P2 + M 3 p 3 ) 2 that is Mipi + M2P2 + M3P3 is also an integral of motion. For the problems of Euler-Grimths and Dubins-Delauney (

0

-Mx

Mn-i\

Mi

M =

0 \M„_i

as we shall see presently. Then,

/

27

/ Mx \ (M,M)

2

= ||M|| , where M = \M„-i/

and ||MP|| 2 = ||M|| 2 p 2 + ( £

Pi+1Mi)

. Hence,

I3 = -||M|| 2 ||F|| 2 + ||M||2p2 + ( $ > + i M i J . With these integrals of motion at our disposal we are now ready for the solutions of our problems. 3.4

Solutions for the problems of Griffiths and Dubins

Let us recall the decomposition of son(ffi) as the direct sum j + p with j isomorphic to so n _i(lR) and p the vector space in which the controls take their values. We shall write M — M\ + M2 for matrices M in so n (E) with Mi € p and M 2 € j . It is an immediate consequence of the Maximum Principle that

H = ±(M1,M1)+Pl is the appropriate Hamiltonian for the problem of minimizing

\f (U(t),U(t))dt over the trajectories of (2.3), and that extremal controls u = (m,..., are given by u = Mi. Hence ft* = Mi, f2p = E\ with / 0 -e 0 • • • 0 \ 1 0 0 Ei =

For the semi-simple case the extremal equations (3.2) become

w n -i)

28

^

= \MUM) + [EUP],

and

^

= [MUP] + [EUM] .

Since [j,p] C p, and [p,p] C j , it follows that

^ = t*.n ^ - » . f = w . i + (*.*]•

M

Therefore, M2(t) = constant. Because of the transversality conditions relative to manifolds Si and 52, the extremal curves l(t) annihilate the tangent vectors of Si and 52 at t = 0 and t = T. Since the tangent spaces of Si and 52 translated from the left to the group identity coincide with the subalgebra j , it follows that M2 = 0. Equations (3.8), written more explicitly, are then given by the following expressions: dMi -jj-=-pi+i(t), dp ~dt dp

t=

l,...,n-l

n-1 1=1

= -Mi(pi

-e),

i=

2,...,n

The reader may verify that the preceding equations are also valid in the Euclidean case. We shall now obtain the remarkable differential equation of Griffiths for the geodesic curvature K,(t). Recall that

J{t)=u\

+ --- + ul_i = Mt + -.- + Ml_i = \\M(t)\\2

Hence, ,

-JTJ

2/c k = 2M • —

71— 1

= - 2 ^2 MiPi+i . i=i

Therefore,

29 (n-\

«2K2= (^Mift+ij

=J3-||M|| 2 P l + ||P||2||M||2

2 = h K^H-\^\{h-e\\Mf)\\M\\

I3-K2{H-±KA

+{I2-EK2)K\

or, equivalently, , 2 _= - .- lKK 44 ++ ((i if f_-££))KK22 ++ (( // 2 -_i if72 2 K ) )++ 4- f 2

(3.10)

This expression coincides with the one obtained by Griffiths in case H = 0 and e = 0, ( 3 , p. 153). The time interval [0,T] is variable in Griffiths' formulation of the problem which explains H = 0 in this paper. We will now show the second remarkable discovery of Griffiths that K2(t)r(t) = constant, where r(t) denotes the torsion of the extremal curve in question. In the process we will also show that each extremal curve is confined to a three dimensional submanifold of M, a fact that was also noticed by Griffiths. To show these facts it will be necessary to recall the Serret-Frenet frames in three-spaces, consisting of the tangent, normal and binormal vectors denoted by t(i),n(t) and b(t) respectively. Their differential equations are the following

*; ^

=

K(i)n(t)

'

= -K(t)t(t)+r(t)h(t),

(3.11)

under the assumption that —— is linearly dependent on t , n, and b . at Since t(t) = x(t) = vi{t), the first leg of the Serret-Frenet frame and of the J-frame coincide. Since the deformations of the J-frame are given by

30

-j±

M

= - ^

i (*)«i+i (*) »

- ^ = Mt-! (t)vi (t)

and

t = 2,..., n

(3.12)

we have,

«(«>*(*) = £ ' = ^ = - ' £ ^ 1 . 8=1

and so +i

i=l

i=l

It then follows from equations (3.9) and (3.12) that dn

"

fi

iVi

« +K-^- = J2p i=2

/n-l

\

M v

=

- \'52 i) i \j=l

n

J2piVi - K 2 * W • i=2

/

Hence, -Dn

1^

-

k_

i=2

It follows from (3.11) that r b = —-—(- nt, which together with the equadt tion above implies that

rfi=^g(ft + ^ - i ) i ; « i=2

Then, 1 n r 2 = ^ ^ ( p ^ + ziM^i) 2 i=2 1 ™ »=2

31

Recall that

KK

T2K4

= — ^ Pi-^i-i, and therefore i=2

= £2 p2n2 — k2K2. i=2

But then

h = - | | M | | 2 | | P | | 2 + \\M\\Vi + (

^PiM^

M=l n

E K pi + K K lZ l ,

- 2i i2

,

i=2

therefore T^V = - J 3 Hence, /t 2 (£)r(t) = constant as claimed. I t rremains to show that —— is in the linear span of the Serret-Frenet triad It dt t, n, b . 2 ..Dri ,-.. „ . D n Since K2rb = n2 ——h K3t it suffices to show that ——— is in the linear dt at* span of the Serret-Frenet triad. To see this consider

kn + K-— = 2_^PiVi — K t dt i=2

Then, B

f

v^ .

j=2

i=2

Dvi

i=2

»=2

= (p\ — e)K,(t)n(t) — kt . 2 VZ

Dn

.

„

Therefore, is in the linear span of t, n and b . Having reduced the solutions to three dimensional problems, the remaining equations can be integrated by quadrature in terms of the geodesic curvature K,(t).

32

We now turn attention to the problem of Dubins. The passage to the extremal curves is straightforward thanks to the Maximum Principle. Nevertheless, there are some minor conceptual differences due to the constraints. The case n = 2 is slightly different because the boundary of u\ < 1 consists of isolated points ± 1 . In all other cases, the boundary is a smooth manifold Sn. Hence, the passage from the planar case to higher dimensional is more varied than for the problem of Griffiths. According to the Maximum Principle the optimal trajectories g(t) generated by the controls U(t) are the projections of integral curves of time varying Hamiltonians H\ of the form ra-l

Hx = -X + Pi +

Y^uiMi i=l

subject to the maximality condition that n—1

-\+Pl{t)

+ ^2ui{t)Mi{t)< i=\

n—1

-\+Pl{t)

+ Y,ViMiit)

>

i=l

for almost all t and for any vector v with v\-\ h v„-i < 1. The constant A is either equal to zero in the abnormal case, or equal to 1 in the regular case. In addition, H = 0 almost everywhere along the extremal curves. All of these conditions are dictated by the Maximum Principle. For time-optimal problems the only difference between the abnormal and regular extremal curves is that the Hamiltonians differ by a constant (i.e., they correspond to different energy levels). All points at which M(t) / 0 on an open interval (to,h) the maximality condition determines a single Hamiltonian H = p\ + \\M\\ with U(t) = —x——. For abnormal extremals H = 0, while for regular extremals

l|M(t)ll

H = \. For n = 2, M(t) is one dimensional, hence, M(t) is either positive or negative, and U(t) = sgnM(t). So on intervals {to,h) on which M(t) ^ 0, the extremal curves are circles (since the curvature is ±1). These circles concatenate at points t at which the curve M(t) crosses the switching surface M = 0 transversally. As is well known, the extremal curves for which M(t) = 0 on an interval project onto geodesies, which essentially explains the first part of the discovery of Dubins that optimal paths in E2 must be concatenations of circles of radius 1 and straight line segments.

33

The second, and perhaps more difficult part of Dubins' discovery, is that optimal arcs cannot contain more than three switchings. We shall not go into these details further. The solutions of Dubins' problem on 2-dimensional surfaces are well understood, and documented in full detail see 12 , 10 , and 15 . Instead, we will proceed to dimensions bigger than 2. Let us first assume that M[t) ^ 0 on an interval (to,h). We shall follow the notations of 13 and use (j,\(t) to denote the quantity ||M(£)|| with A denoting the dependence on the multiplier A that differentiates regular curves from abnormal curves. Otherwise we will follow the notations used in the Euler-Griffiths problem. The equations (3.3) and (3.5) are applicable. For e ^ 0 the extremal dM dP equations are — - = [n*,M] + [Q,P,P], and — = [fifc,-P] + [fi P) M] where ftk and Clp denote the appropriate projections of dH corresponding to H = pi + ||Mi ||. Therefore, fifc

= 177T7, = — M i ' a n d ttP = Ei . ||Mi|| MA As in the problem of Euler-Griffiths the projection of M on so„_i(R) is zero, and hence M = Mi. Therefore, ^

at

= [EltP] ,

and

^ = —[MUP] at /j,\

+ [E^M,]

.

The preceding equations can be written more explicitly in terms of the coordinates M i , . . . , M„_i of M as follows: dM; =-Pi+i dt

a}

P+ dt

l,...,n-l,

ra-1

dJ d i 1

i=

= ~Y,M^

and

1=1

** fP1 = - M i \fix

e

i - i i=

therefore -l

A«AAA =

-"Y^Pi+iMi. i=l

1 l,...,n-l

34

Now we can proceed much like in the problem of Euler-Griffiths: Since H^n

+ px,

J 2 = ||P|| 2 +

e/4

we have (MAAA)2

i 2 | | | | JI> ||22 , II JCT | | 22„ 2 = h - II^IIIIMiH + UMill ^

= h-A(H-&?

+ (I2-evl)til ^A 2

Therefore, (i\ = -(l + e)iJ,l + 2Hfix + I2-H2

+^

(3.13)

^A

where H = 0 for A = 0 and H = 1 for A = 1. Recall that equation (3.13) is derived under the assumption that M{t) ^ 0 on an interval. This assumption implies that the curvature of the associated extremal curve is identically equal to 1 in this interval. Hence, (3.13) contains the solutions for the problem of Delauney.This equation agrees with that of 13 , but differs from the one obtained by Griffiths on 3 p. 156 for the problem of Delauney. (Griffiths' variable A seems to be the same as the variable pi\ in this paper). We shall now show that the torsion r(t) of each extremal curve satisfies the relation Tfi\(t) — constant. Again we draw from the calculations obtained in the Euler-Grifnths case. It follows that the frame deformations in this problem differ from those in the problem of Euler-Griffiths by a factor fi\ i.e., Dvi —j— = at

1 ^ > MiVi fi\ *—'

and

Dvi 1 —j— = — M t v i , at n\

.

i=

2,...,n.

Since the curvature is equal to 1 in the interval (to, £i), it follows that the normal vector n(t) is given by n(t) =

^ MM, or H\ti(t) = - J2 MM. MA i=2

j=2

After the calculations analogous to the previous problem, with n replaced by fJ-\, we get that

(MAT)2

= YlPivl «=2

hence, (fj,\(t)) r(t) = constant.

~ PAAA = ~Is '

35

We shall omit the calculations which show that the derivative —r- of the dt binormal vector is in the linear span of the triad t, n, b , since they are the same as before. Consequently, we have recovered the result of D. Mittenhuber that each extremal curve is confined to a three dimensional submanifold of the ambient manifolds n . Of course, geodesic curves must also be the solutions to the problem of Dubins, hence there are solutions for which ||M(£)|| = 0. The algebraic surface S given by Mi = M 2 = • • • = M„_i = 0 is called the switching surface for this problem. We shall not go into any detailed analysis of the type of crossings of S, except to quote the following theorem 13 : A curve of minimal length for the problem of Dubins is necessarily one of the following: a.

A smooth arc with constant curvature equal to 1.

b.

A geodesic arc.

c.

A continuously differentiable curve that is a concatenation of geodesic arcs and arcs with curvature equal to 1.

It is still an open (but likely difficult) problem to determine the maximal number of switchings for an optimal curve. As already noticed in 3 any variational problem that involves functionals which depend on the first curvature only exhibits striking similarities to the preceding problems, although this observation has not been elaborated in full details yet. We now shift attention to the elastic problems. 3.5

The Elastic Problem and the Mechanical Tops

Recall that the elastic problem is defined through a positive definite quadratic form ( , ) on so„(E), and the parameters czi,... , a n . The Hamiltonian equations for this problem are most naturally expressed with the aid of the trace form (A,B)

= — -TrAB.

To begin with, the trace form is positive definite

on so n (M), and therefore (U, U) = (U, JU) for some positive definite (relative to the trace form) linear mapping J : so„(R) —> son(R). Each of the Lie algebra g in this paper is of the form g = 6 © p with p an n-dimensional vector space. We shall adhere with the notations established earlier and write

36

( 0 -pi epi

- p2

:

pn\

= P +M

M

\epn

/

for an arbitrary element L in Q with e ^ 0. In the Euclidean case /0

0--0\

Pi

L =

:

M

\pn

J

With these notations at our disposal, the Hamiltonian associated with an elastic problem is given by n

1 1

H=-(M,J- M) The parameters ai,...,an

+ Y,aiPi-

can be identified with a matrix A in Q given by ( 0 -eai

A=

ean\ 0

\an in which case (A, P) = ^ atPi, therefore the preceding Hamiltonian H can be also written as

H=-(M,J-1M)

+ (A,P) .

We remind the reader that this Hamiltonian is obtained through the maximally condition of the Maximum Principle. The maximum of -±(U,JU)

+ (A,P) + {U,M)

relative to the variables U occurs at U = J

M, and is equal to

^(M,J-1M) Then dH = A + J_1M. cases are ^

+ (A,P).

The extremal equations in the non-Euclidean

= [J-*M, M] + [A,P],

*£ = [J-'M, P] + [A, M)

whereas in the Euclidean case we have ^ - = [J-1M,M] +

PAA,

dp = dt

(-J~1M)P

(3.14)

These equations bear striking resemblance to the equations of motion of a rigid body under the influence of the gravitational force. To explain these somewhat mysterious connections we shall restrict our considerations to the Euclidean case and begin with the simplest case, the planar pendulum. Assume that a bob of mass m is attached to the pendulum of length I which swings under the gravitational force F = mge\. We have oriented the Euclidean plane by the right hand orthonormal frame ei, e-i that is centered at the fulcrum of the pendulum. We shall denote by (ai(£), 02(f)) the orthogonal frame attached to the pendulum and so adapted that the position q(t) of the free end is in the direction of Si(t) i.e., that q(t) = lai(t). See figure 1. Let R(t) denote the isometry defined by R{t)ei = Si(t),i = 1,2. Then each movement of the pendulum traces a curve R(t) in 502(M) given by q(t) = lai(t) = lR(t)ei. The kinetic energy T associated with each movement q(t) is given by dq(t) dt

- 2 ^

dR. ei dt

Since R(t) is a curve in S 0 2 ( dR(t) _

dt

R(t,

( 0

~H[T){w(t)

-w(t)

0

38

Figure 1. The pendulum

for some function w(t). The kinetic energy expressed in terms of w(t) is given by

T=]-ml2w2{t). The potential energy V in raising the pendulum from its equilibrium position q(0) = i€\ to an arbitrary position q(t) is given by

= mgl(l - (ei,.R(*)ei)) , where ( , ) denotes the Euclidean metric on E 2 . Therefore, the Lagrangian L — T — V is given by L(w,R) = -ml2w2

+ (ei,R(t)ei)

— mgl,

which we interpret as a function on the tangent bundle of S02(M) realized as the product so 2 (K) x S02(IR). According to the Principle of Least Action the movements of the pendulum minimize the integral / Jo

L{w(T),R(T))dr

39 over all paths *R(t) dt

( 0 = R{t) -^>\w(t)

"»(*) 0

subject to the appropriate boundary conditions. The Maximum Principle immediately yields the Hamiltonian H on T*S0 2 (E) of the form = ^pM2

H(M,R)

- mglieuRei)

+ mgl

where M denotes the Hamiltonian lift of the left-invariant vector field

i.e., M(p) = pA\, where A\ = I

1. In addition, the Maximum Prin-

ciple provides a relation between the angular velocity w(t) and the angular momentum M with w(t) =

—-rM(p(t)). ml2. Then, the integral curves of H satisfy

M(t) ml2

™'2

,

ft(A)=mgl(e1,RAe1)

(3.15)

)

for any A in 502(E), as can be verified through the formulas established in 5 . (Note that the above Hamiltonian is neither left nor right invariant.) To recognize these equations in their familiar form parameterize R(t) by an angle 6(t) defined by

n{>

/cos0(i)-sin0(i)\ ~ \sm6{t) cos6(t) J

Then, M

d_R-RH\

°

dt- \MifiL ml2

(t)\ -^nP\

r c ducc, tto redUCeS 0

n

J

*° -

M{t)

dt " mP

40

and

dt

= -77(^1) = m0/(ei,.R(i),4iei) = dt

-mglsinO

Hence, d20 _ 1 d M _ _mgl . dt2 ~ ml2 dt _ ~ r f S i n

'

or g

+

f s i „ » = 0.

(3.16)

While normally investigations of the planar pendulum end with equation (3.16), equation (3.15), which is hardly every noticed in the literature on mechanics, shows connections with the semi-direct product as follows. Let h{t) = mglR~1(t)ei, and let hi(t), h,2(t) denote its components. Then,

dh dt

= - " » < U ~ " o ( t ) ) *-'<<>«• =

and —— = (h(t),A\ei) dt Hamiltonian

H(huh2,M)

= h^U) constitute the equations of the left-invariant

= 7TV 2mt 2

M 2

+ ei> = T ^ 7J2 M 2 2m/

+ /l

i

on the semi-direct product G = E2 x S02(M). Of course, the mathematical pendulum is confined to the momentum level h\ + h\ = (mgl)2. So, the movements of the pendulum are subordinated to the problem of the elasticae. The same applies to the spherical pendulum but with minor modifications. The orthonormal frames are taken in the Euclidean space E 3 , and the moving frame a\(t), a2 (t), S3 (t) is adapted to the pendulum so that the position of the free end q(t) is given by q(t) = lai(t). Then each path of the pendulum traces a path R(t) on SOa(]R) defined by di(t) — R(t)ei i = 1,2,3. As usual, each path R(t) generates angular velocities w\(t), ui2(t), W3(t) denned by

41

0 dR(t) = R(t) | w3{t) dt ,-w2{t)

-w3(t) 0 (t) Wl

w2{t) -wi(t) 0

Since the pendulum does not rotate about its own axis we will impose the constraint that wi(t) = 0. That is, we will trace the movements of the pendulum by J-framed curves on S03(K). Then, the kinetic energy T is given by —W3 Wi

0

= ±-ml2 R(t) 2

T=-m

w3 -w2

0 0

0 0

ei

= -ml2{wl+wl) , while the potential energy V retains the same form as in the planar case and is given by V = mgl{l — (ei,i?ei)). The Principle of Least Action for

V)dt Jo

yields the Hamiltonian

H =

2ml2

{Ml + Ml) - mglieuRex)

+ mgl,

where M2(p) = p{A2),M3(p) = p(A3) for all p in so^R) with Ai,A2,A3 denoting the standard basis in so 3 (E). As a consequence of the Maximum Principle, the extremal angular velocities w2(t) and w3(t) are given by the usual relations w2(t) = ——M2(p(t))

and w3(t) =

TTvl

In a manner analogous to the one used in extremal equations dR = RQ{t) , ~dl with

——M3(p(t)). Tflil

and

5

(pp. 376-378) we get the

dM = M xQ, + P{t) x ei dt

42

/

o

M3(t) ml2 , M2(t) ml2 0 -M3 M(t) = ( Af3 0 , - M 2 Mi

n(t) =

M3(t) ml2

(3.17)

0 0

M2 -Mi 0

M(t) = ( M i , M 2 , M 3 ) T and P(f) = m ^ P " After identifying vectors A = (ai,a2,a3)T

(

M2(t)\ ml2

1

^. with antisymmetric matrices

0 -a3 a3 a3 0 — ai —a,2 ai 0

we can rewrite the extremal equations in the Lie bracket form: dR

„.,„,,

dM

= [Sl(t),M(t)]

+ [A1,P(t)]

.

The extension to the semi-direct product is straightforward: Each movement q(t) of the pendulum defines a curve g(t) in the group of 1 0 motions where g(t) = Since P(t) = mglR ei, lq(t) R(t) dP = P(t) x Cl(t) , dt Together with the equations ^^[n,M}

or

dP dt

= im,p(t)}-

+ [A1,p]

these equations coincide with equations (3.14). The equations of motion for arbitrary rigid body in E 3 are done in 5 , and for an n-dimensional rigid body in 14 . For convenience to the reader we shall quickly outline the procedure. Let e i , . . . , e„ denote an absolute frame, and let ai (£),..., an(t) denote a frame attached to the body with both frames centered at a fixed point O of the body. We may orient the absolute frame so that the gravitational force is of the form F = Ce\ where C is a gravitational constant.

43

Denote by R{t) the isometry that relates the moving frame to the absolute frame. If q = (qi,... ,qn)T denotes the coordinate vector of a point OP in the absolute space, and if Q = (Qi,... ,Qn)T denotes the coordinate vector of the same point relative to the moving frame then q = RQ . For each movement of the body a point q(t) on the body undergoes velocity —- = -r-Q. The kinetic energy of each such point is given by at at dq ~dt

1 2m

2

=

1

^m\\u(t)R(t)Q\\2

where u(t) denotes the antisymmetric matrix defined by dR(t) —^- =

,.. . . u(t)R(t).

The total kinetic energy is the aggregate of the contributions due to point masses and can be expressed as an n-dimensional integral in terms of the mass density p. Then, \\u(t)R(t)Q\\2p(Q)dQ

T=\f

L

* ./Body 1

Let U{t) = R~ (t)u(t)R(t).

\\R-1u(t)R(t)Q\\2p(Q)dQ

= \f ./Body

Then each motion of the body corresponds

to a curve R{t) on SOn(R) where — = u(t)R(t) = R(t)U(t). dt kinetic energy is given by

\ [ z

.

The associated

\\U(t)Qfp(Q)dQ

./Body ody

We now express kinetic energy in terms of the quadratic form {UuU2) = \f 1

(U1Q,U2Q)P(Q)dQ, ./Body

where (UiQ,U2Q) denotes the Euclidean inner product. The form ( , ) is positive definite on the space of n x n antisymmetric matrices, and T(U) = (U,U). Thus, the geometry of each rigid body defines its own left-invariant metric on SO„(E). To find the expression for the potential energy it is convenient to introduce the center of mass Q0 defined by

44

<9o /

p(Q)dQ = J

/Body •/Body

QP(Q)dQ.

./Body -/Body

Then the potential energy is V is given by

V = -C J J

/ e i • ™QSp(Q)dQdT = -Cm(euR{t)Q0) ,

where m stands of the total mass of the body. To get the equations of motion we shall assume that the Principle of Least Action is valid. That is, we shall assume that the motions of the body minimize

J (T-V)dr

=J

^(U(T),U(r))-Cm(e1,R(T)Q0)dT

over all curves R(t) in SO n (E) that are solutions of ^ ^ = R(t)U(t). at As in the case of elastic problems, the quadratic form ( , ) induced by kinetic energy can be expressed in terms of the trace form on so n (R), in which case, the Hamiltonian corresponding to the total energy of the body is given by H=^-(M, H

J~XM) + Cm{R-leuQ0)

.

To make the connection with equations (3.14) rename the variables P(t) = m C R - 1 e i , Qo = a, and identify

x(t) = / R{T)etdT Jo and R(t) with the matrix

»<*> = (*(t) R°(t)) Then,

45

/oo-

• o\

1 = 9(t) 0 il(t) dt

V

dM dt

[n(t),M(t)]

+ [A,P(t)},

7

and dp

= m),P(t)}

where ( P(t)

/o o

0 0 Pi(t) 0

0

\pn(*)0

•••0/

and

A =

0

ai 0 Van°

•••

°/

The preceding developments reveal a remarkable and unexpected phenomenon: The movements of an n— dimensional top always form an invariant subsystem of an elastic problem on a Euclidean space E™ For the top the extension to the semi-direct product is redundant. Indeed, once the angular momentum matrix M(t) is determined, the angular velocity U(t) is given by U(t) — J~lM(t), and hence the rotation matrix R(t), as the solution of the matrix differential equation —— = R(t)U(t),

is completely

determined. The translational momentum variable P(t), given by P(t) = CmR~1(t)ei, corresponds to the body coordinates of the fixed vector Cme\ and is determined by the rotation R(t). The same is not true for the elastic problems, even though R(t) and P(t) are intertwined by the conservation law R(t)P(t) — constant. Hence, the top is an invariant subsystem of the elastic problem denned by R(t)P(t) = Cme\. The central line of the top x(t) defined by x(t) = x0 + / R(s)eids, Jo which is never mentioned in the literature on mechanics, carries the geometric information which links the top to the geometric problems considered in this paper. To be more precise about this point we shall now focus on three dimensional tops, called heavy tops.

46

3.6

The top of Lagrange and the problem of Euler- Griffiths

For three dimensional problems the deformations of frames are described by K = SOs(]R), and the elastic quadratic form is a quadratic form on the Lie algebra k = 503(E). We shall continue with our conventions that matrix

(

0 - a 3 a2 \ a3 0 - a i -a2 ai 0 J

is identified with a vector A = (aia2,a,3)T. Then, [A, B] = B x A where x denotes the cross product in E 3 (Remember that [A,B] — BA — AB). Furthermore, (A,B) = A-B with A • B denoting the Euclidean metric in E 3 . We shall use Aj, A2, A3 to denote the eigenvalues of the form ( , ) relative to the trace form, called the principal moments of inertia in the literature on mechanics. Let Ai, A2, A3 denote an orthonormal basis in 503 (K) relative to the trace form of eigenvectors of J. We have that (A, A) = X1al + X2al + \za\ ,

for any matrix

A — o,\A\ + a2A2 + a3A^ .

Furthermore, we may assume that Ai, A2, A3 are positively oriented in the sense that i i x A2 = A3 ,

A3xAi=A2,

A2xA3=Ai,

or equivalently, we may assume that [AUA2] = -A3

,

[A1,A3] = A2,

and

[A2,A3] = -A1

.

For elastic problems on three dimensional space forms the Lie algebra of G is six dimensional, so we need to extend this basis in 503(E) to all of g. The most natural way to extend this basis is to define

Bl

/ 0 -an -U 0 ) '

(0 -eAf\ (0 S '{A2 0 ) ' * -U

B2

-SAT\

0 ) •

Then Ai, A2, A3, Bx, B2, B3 is an orthonormal basis on g relative to the trace form. The commuting relations for the preceding basis are given by table 1, as can be easily verified

47

Table 1. A2 ~A3

A! At A2 A3 Bx B2 B3

0

A3 A2 -Ax

0

^3

-A2

0

Bx

0

B3 -B2

0

B3 B2 -Bx

B2 -B3

0

0

A! -B3

B2

0

B1 -eA3

0

-Bi

eA3

0

eA2 -eAi

Bx

0

-EA2

eAi

0

B3 -B2

3

3

As before, we shall write L = M+P with M = £ MiAt and P = X] Pi#»1=1

8=1

In these coordinates the Hamiltonian H = — (M, JM) + (A, P) is equal to

2 \ Ai

A2

+ aipi + a2p2 + a3p3

A3

where a\, a2, a3 denote the coordinates of A with relative to Bi, B2, B3. Mi Mn M3 Then dH = -r^-Ax + -r~A2 +-r1 + axBx + a2B2 + a3B2 and the HamilAi A2 A3 tonian equations (3.3) and (3.5) expressed in terms of the vector products on R3 are as follows: dM

A

A

1

where M = (MxM2,M3)T,P

dP

A

A

, ,~

Is

r (3.18) — = P xtt + e(M x A) dt 1 1 1 = (Px,p2,p3)T,(l = (—M 1 ; — M 2 , — M 3 ) T

—— = M xQ + Px A , at

Ax

A2

A3

and A = {ax,a2,a3)T Equations (3.18) coincide with the fundamental equations in 5 except for the minor changes in notation. It is known, although not yet published, that two eigenvalues among Ai, A2, A3 must be equal to each other for (3.18) to be completely integrable. In the case that two eigenvalues are equal to each other, there are essentially only two cases that are completely integrable: the case of Lagrange and the case of Kowalewski 6 . The case of Kowalewski, considerably more challenging to understand and explain, will be treated in another paper. Here we focus on the case of Lagrange and relate its solutions to the solutions of Euler-Grifnths problem obtained earlier.

48

For Lagrange's elastic problem we assume that A2 = A3 and that a 2 = a3 = 0. Then it follows immediately from equations (3.18) that M\(t) is constant. Since a\ + a\ + a3 = 1 and a2 = a3 = 0, it follows that ai = ± 1 . We shall take a\ = 1 (The other choice ai = — 1 is isomorphic to this one). The control system that gives rise to this Hamiltonian is of the form: / 0 -e 0 0 \ 1 0 -u3 u2 0 «3 0 —«i

dg

di=9

\0 -u2

ui

0 J

dx If x(t) denotes the central line then x(t) = g(t)ei and — = <7(£)e2. Moreover, dt Ddx , , D_dx_ 2 -r—r- = uzge-z—U2gei, and therefore — (wj+wl). Thus, the geodesic at at ~dt~dl curvature n(t) associated with the central line x(t) satisfies K2 = u\ +u\, and hence

-J

{u,u)dt= ±J {K2+u\)dt.

Since the extremal controls are given by ui(t) = T-Mi(i) , Ai

u2{t) = -M2(t) A

,

u 3 (t) =

A

TM3(t),

with A denoting the common value A2 = A3, it follows that Ui(t) = constant and that M2 + M2 = \K2 . In particular when A = 1, then the geodesic curvature of the elastic curve x(t) is equal to y/M% + M | . Hence the Hamiltonian H corresponding to Lagrange case can be rewritten as

H

=r2

+

i-1Mi+pi

1 or in the reduced form as HQ = -K2 +pi, since M\ = constant. The equation

US) 2 - ( M * - 5^)" -.(. + »?))-(«.- (H- i«)«.)2 obtained on p. 460 in 5 can be easily translated into the notations in this paper. The constant of motion M\ = Hihi + H2h,2 + H3h3 is related to the

49

integral of motion I3 in this paper by Mi = ^fh. The other integral of motion \\h\\2 +e\\H\\2 is the same as J2 in this paper. Moreover, v2 = K2. Hence, the above equation becomes

K2k2 = K2 (l2 - (H0 - \K2)2 It was also shown in is given by 2 K

4

- S(K2 + M2)) - (-y/h-

(Ho -

^

2

)^i)

that the associated torsion r(t) of the central line

(t)r(t)

= Hi (H0 + i « 2 ) - Mi ,

which rephrased in the notations of this paper becomes

K2T

= Mi

Thus K2T = constant whenever Mi(t) = 0. Then K2K2 = K2 (h -{H-

\K2)2

- eA

+ h •

Remarkably, the preceding differential equation for K coincides with the equations (3.10) for the problem of Euler-GrifBths. We may summarize some of our findings in the following way: The elastic curves for the Euler-Grimths problem for n > 2 coincide with the elastic curves for the Lagrange's elastic problem whenever the latter are subjected to the same boundary conditions (Mi = 0). We may also rephrase the preceding observation by saying that the top of Lagrange generates a central line which is an elastic curve. More specifically we may relate the elastic curves of Euler-Grifnths to the movements of the spherical pendulum via the equation (3.18). In this context, the position q(t) of the pendulum dx and the corresponding elastic curve x(t) are related by — = q(t). Thus, the pendulum traces the tangent line of the elastic curve. This observation fits nicely with 4 which relates the planar elasticae to the movements of the planar pendulum. The analogy with the top further illuminates the significance of elastic curves. As we have just seen the elastic problem naturally extends to all space forms, while the top does not (at least not in some convincing way).

50

As a result, the elastic problems reorient our understanding of completely integrable Hamiltonian systems associated with the tops and suggest natural extensions to arbitrary Lie groups. Among many compelling and intriguing questions that can be asked about elastic curves, we shall single out the ones pertaining to closed elastic curves. Langer and Singer in 7 showed that all possible toral knots occur as the solutions of Euler-Grifnths problem for the Euclidean case. In view of the fact that the Euler-Grifnths problem leads to a completely integrable Hamiltonian system, the conclusions of Langer and Singer are not surprising. However, could the result of Langer and Singer be made more transparent through suitable action-angle arguments? More generally, does the same conclusion hold for the elastic curves on non-Euclidean spaces. Finally, passing to the case of Lagrange, we may ask for the types of knotted curves which occur as the solutions of the elastic problem for which K2T is not necessarily constant. References 1. C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, (Reprinted 1982, Chelsea Publishing Co., New York, Berlin, Teubner (1935)). 2. L.E. Dubins, On curves of minimal length with a constraint on average curvature and with prescribed initial positions and tangents, Amer. J. Math, 79, 497-516 (1957). 3. P. Griffiths, Exterior Differential Systems and the Calculus of Variations, (Birkhauser, Boston, 1983). 4. V. Jurdjevic, Non-Euclidean Elastica, Amer. J. Math., 117, 93-125 (1995). 5. V. Jurdjevic, Geometric Control Theory, (Advanced Studies in Mathematics, 52, Cambridge Univ. Press 1997). 6. V. Jurdjevic, Hamiltonian Systems on Lie Groups: Kowalewski type, Annals of Math,150, 1-40 (1999). 7. J. Langer and D. Singer, Knotted elastic curves in R3, J. London Math. Soc, 2, No. 30, 512-534 (1984). 8. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, (4th Edition, Dover, New York 1927). 9. D. Mittenhuber, Dubins' Problem in hyperbolic space, in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 101-115 (1998). 10. D. Mittenhuber, Dubins' Problem in hyperbolic plane using the open disk

51

11. 12. 13.

14. 15.

model in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 115-153 (1998). D. Mittenhuber, Dubins' Problem is intrinsically three-dimensional, ESAIM: Controle, Optimisation et Calcul des Variations, 3,1-22 (1998). F. Monroy-Perez, Non-Euclidean Dubins problem, A Control Theoretical Approach, (Ph.D. Thesis, University of Toronto, 1995). F. Monroy-Perez, Three dimensional non-Euclidean Dubins' Problem in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 153-183 (1998). T. Ratiu, The Motion of the Free n-Dimensional Rigid Body, Indiana Univ. Math. J., 29, No. 4, 602-629 (1980). H. Sussmann, Shortest 3-Dimensional Paths with a Prescribed Curvature Bound, Proceedings of the IEE Conference on Decision and Control, 33063312 (1995).

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53

CONTROLLABILITY OF LIE SYSTEMS J. D. LAWSON Deparment of Mathematics of Louisiana State University E-mail: [email protected] D. MITTENHUBER FB Mathematik, AG 5, TV Darmstadt, Darmstadt Germany E-mail: [email protected] The purpose of this paper is to present conditions that allow one to determine the controllability of a Lie system on a connected Lie group G by pushing the controllability question forward to smaller dimensional coset spaces. Since noncontrollable systems may become controllable on smaller dimensional coset spaces, the problem considered is the determination of closed subgroups H such that a system is controllable on G/H if and only if it is controllable on G.

1

Introduction

In recent years there has been an increasing interest in geometric control problems that can be formulated on Lie groups with control set a subset of either the left or the right invariant vector fields. For example, V. Jurdjevic has successfully treated various aspects of certain classical problems in mechanics by interpreting them in the context of control problems on Lie groups (see, for example, Chapter 6 of 5 ) . The advantages of interpreting problems in this framework include having available both the machinery of geometric control theory and the rich arsenal of highly developed theoretical tools from the theory of Lie groups and Lie algebras. The primary analysis in such problems generally involves application of the Maximum Principle, but one typically needs to check that the system is controllable at the initial stage of the investigation. The primary aim of this paper is to present some tools that are helpful for the determination of controllability for systems on Lie groups. The main technique involves pushing the controllability question forward to a potentially easier one on a smaller dimensional coset space. In this context one seeks to be able to recognize those subgroups H of a Lie group G with the property that a system is controllable on G/H if and only if it is controllable on G. For the analysis of such problems we find the consideration of subsemigroups of Lie groups with non-empty interior to be a useful tool. In the first sections we introduce and develop the basic concepts with which we work. The main focus of this paper is the setting of solvable Lie groups and we consider those in some detail in the main sections of the pa-

54

per. We close with brief remarks concerning the work of L. San Martin and collaborators in the semisimple setting. 2

Control systems on Lie groups

Let G denote a Lie group with Lie algebra g (one may think matrix Lie groups, the primary example, although the results of this section are valid for more general Lie groups). We henceforth identify g with the set of right invariant vector fields on G. For X 6 g and g € G, let X{g) € TgG denote the value of the vector field X in the tangent space TgG at g. Let M be a smooth (= C°°) manifold and suppose that $ : G x M -»• M is a smooth action of the Lie group G on M. We typically denote $(g,x) by gx or g.x. Let V°°{M) denote the Lie algebra of smooth vector fields on M. For x S M, the smooth mapping <J>X : G -> M given by §x{g) = g.x has derivative at e, c ^ : TeG -> TXM; alternately d$x(v), v G TeG, is given by » 4 G is the exponential mapping and X(e) = v. The mappings d$x give rise to a Lie algebra homomorphism d$ : g —> V°°(M) given by <M>(X)(x) = d$ x (X(e)) (note that the appropropriate match-up to obtain a Lie algebra homomorphism is right invariant vector fields with left actions). We denote the vector field d$(X) by X. Reference 21 provides a detailed treatment of transformation Lie groups. Let SI be a subset of 0. We consider the Basic Group Control System (BGCS) on G given by the differential equation g(t) = U(t) (g(t)),

(BGCS)

where U(-) is called a steering or control function and belongs to the class U(ft) of measurable functions from R+ = [0,co) into SI which are locally bounded, that is, bounded on every finite subinterval. (A measurable set in g is one whose image is measurable in the euclidean space which is vector space isomorphic to g; equivalently the set is measurable with respect to Haar measure on (g, +).) In the case of a matrix group G with matrix Lie algebra g, the Basic Group Control System becomes the control matrix differential equation g(t) = U(t) • g(t). A solution of (BGCS), called a trajectory, is an absolutely continuous function x(-) from E + into G such that the equation (BGCS) holds a.e. Control systems such as the one just described are called right invariant, since right translates of solutions of (BGCS) are again solutions. Lemma 2.1. Let U : R + —> ft C g be locally bounded and measurable. Then there exists a unique solution x(-) on G of the initial value problem x(t) = U(t)(x(t)),

x(0)=g.

55

This solution is globally defined in the sense that x(t) is defined for all t > 0. Proof. By passing to a chart, one obtains a system of ordinary differential equations satisfying the Caratheodory conditions; hence there exist unique local solutions. The homogeneity of G is used for extending local solutions to global solutions. See Lemma 2.1 of 6 for details. • If U is a locally bounded measurable steering function, we denote the solution x(-) of (BGCS) which satisfies x(0) = g by n(g, -,U), i.e., 7r(g,0, U) — g and 7r(<7,£, U) = x(t) for all t > 0. If h = ir(g,t,U) for some t > 0, we say that U steers g to h in t units of time. If there exists U £ U(Q) such that h = Tr(g,t,U), then we say that h is attainable from g at time t for the system SI. The set of all such elements attainable from g at time t is denoted A(g,t,Q). We also employ the notation

A(g,
(J

A(g,t,0),

0
A( fl ,fi)=

(J

A(g,t,n).

0
The set A(g,Cl) is called the attainable set from g. We use the subscript "pc" to denote the appropriately modified versions of the preceding notions for the case that only piecewise constant controls into ft are used. For example, the set of elements attainable from g at time t using only piecewise constant controls into fi is denoted Apc (g,t,il). Note that the equality A(g, Q) = A(e, Cl)g and analogs for other attainable sets follow from the right invariance of the control system. Thus attainable sets at the identity determine the attainable sets at other points, and we can restrict our attention primarily to the former. The following theorem is a summary of results surveyed in n . Theorem 2.2. Let 0 ^ tl C g. (i) A p c ( e , n ) C A(e,H) are both subsemigroups of (exph^), the subgroup generated by exp(hfj), where h n is the subalgebra of g generated by fL (ii) The subsemigroup Apc(e, Q.) is equal to the subsemigroup generated by exp(fi). (hi) The semigroups Apc(e,£l)

and A(e, Q) of G have the same closure in G.

(iv) The semigroups Apc(e,il) and A(e, J7) have non-empty interior if and only if Si generates (as a subalgebra) g. In this case the interior of each semigroup is dense in that semigroup.

56

(v) IfS = A(e,fl)~,

then

L(5) := {X eg: exp(tX) £ S for all

t>0},

called the tangent wedge of S, is a closed convex cone, and Jj(S) fl — L(5) is a subalgebra of g. Furthermore, h(S) is the largest subset of g having attainable set contained in A(e,fl)~. That fl generates g is the "full-rank" condition in the Lie group context. This motivates our focus on semigroups with non-empty interior. It is convenient to also denote the semigroup A(e, fl) by S(fl) and Apc(e, fl) by Spc(fl). Note that since the attainable or reachable set for g 6 G is S(fl)g, it follows that fl is controllable on G if and only if S(fl) = G. Since a dense subsemigroup with non-empty interior of a connected group G must be G (see 7 ) , we have that if 5(f)) is dense, then Spc(fl) = S(fl) = G, and the system is controllable and piecewise-constant controllable. These control theoretic notions on G have analogues for a smooth Gtransformation group $ : G x M -t M. For fl C g, we consider the basic manifold control system (BMCS) on M given by the control differential equation x(t) = U(t)

(x(t)),

where £/(•) : K+ -> fl is locally bounded and U(t) = d$ (U(t)). Proposition 2.3. The solution to the (BMCS) ±(t) = U(t) {x(t)),

x(0) = x

on M is given by x(t) = g{t).x, where g(-) is the solution of (BGCS) with steering function U(-) and initial condition e. The basic control differential equation on M has a global solution for any initial value and A(x, fl) = S(fl).x for all x € M. Proof. The first assertion follows from x(t) = d^xg(t)

= d$x (U(t)(g(t))) =

= d* f l ( 0 . I (l/(t)e) = U(t)(g(t).x)

d$x°dpg{t)(U(t)e) = U (x(t)),

where pg (h) = hg is right translation in G. That global solutions exist now follows from the corresponding assertion for the (BGCS). Also the last assertion follows readily from the first. •

57

3

Groups irrelevant for transitivity

Prom the preceding section we have the following general principle for controllability of invariant systems on Lie groups: A subset il of the Lie algebra g of a connected Lie group G determines a controllable system on G if and only if the attainable set at the identity e is all of G, i.e., if and only if S(fi) = A(e, CI) = G. It can be a quite difficult problem to determine whether or not this is the case. Sometimes the recently developed Lie theory of semigroups (as presented in 1 or 2 , for example) can be helpful in this regard. In this paper we consider the strategy of reducing the problem to a lower dimensional, hopefully simpler, problem. Homogeneous spaces are often good candidates for such reduction. Suppose that a topological group G acts continuously on a space M. The space M is a homogeneous space if the action is transitive (G.x = M for all x £ M) and if further the mapping g i-> gx : G —> M is an open mapping for each x £ M. Let S be a subsemigroup with non-empty interior (a general setting that includes the case of the A(e, 0) for fi generating). Let M be a homogeneous space for an action of G. We say that the semigroup S acts transitively on M if Sm = M for all m £ M. We would like to know conditions on G, M, and perhaps S that ensure that S = G if 5 acts transitively on M. We recall a result of San Martin and Tonelli 25 . Proposition 3.1. Let G be a topological group acting on the homogeneous space M. Let S C G be a subsemigroup with nonempty interior. For p £ M, let Gp denote its stabilizer subgroup. If S acts transitively on M, then Gp D int(S) ^ 0 for all p £ M. Conversely if M is connected and Gp D int(S) jt 0 for all p 6 M, then S acts transitively on M. Proof. Let p 6 M, let s0 € int(5), and let q = sQp. If S is transitive on M, there exists t e S such that tq = p. Then ts0 £ S (int(S)) C int(S') (since S (int(S)) is an open subset of S), and (ts0)p = p, i.e., tsQ 6 int(S) f~l Gp. Conversely suppose that p 6 (intS)p for all p e M. Let T := int(S). Then T and T _ 1 are open subsemigroups of G, p € Tp for all p e M, and hence p £ T~lp for all p £ T. Let x £ M, let U = Tx and let V = U{ T _ 1 2/ : V & U}. Then U and V are open since M i s a homogeneous space, and U U V = M since y £ T~ly. If U D V ^ 0, then there exists sx = t~xy for some s,t £ T and y £ U. Thus y = tsx £ Sx = U, a contradiction. Hence U f~l V = 0. We conclude fromn connectivity that V = 0, i.e., that Tx = M. Thus S acts transitively. • Corollary 3.2. Let G be a connected group acting on the homogeneous space

58

M and let S C G be a subsemigroup with nonempty interior. If S acts transitively on M and if Gp is compact for some p € M, then S = G. Proof. By transitivity of the action M = Gp is connected. By the previous proposition there exists s e Gp n int(5) ^ 0. Pick U = U'1 open containing the identity e of G such that sll C int(S). Since Gp is compact, we can find n such that sn € U~\ hence e G snU = sn-1sll C 5 (int(S)) C int(S). Since G is connected and int(5) is an open subsemigroup containing e, it follows that int(S) is all of G, and hence S = G. • For the following definition, we have in mind closed subgroups H and the corresponding homogeneous spaces G/H, but the definition applies to arbitrary subgroups H and the corresponding set-theoretic action of G on the cosets G/H. Definition 3.3. A subgroup H < G is irrelevant for transitivity (IFT) if for every subsemigroup S C G with int(S) ^ 0 the conclusion: S transitive on G/H (S.x = G/H for all x £ G/H) = • S = G holds true. If a group H is not (IFT), we say that it is (NIFT). Corollary 3.4. Let G be a connected Lie group and let H be a compact subgroup. Then H is irrelevant for transitivity. Proof. Suppose that 5 is a subsemigroup with non-empty interior. If 5 acts transitively on G/H, then by Corollary 3.2, S = G, since the isotropy group at e = eH = H is H. D Let G be a connected Lie group and let H be a closed subgroup. Then the homogeneous space M := G/H admits a (unique) smooth structure such that the action $ : G x M -> M is smooth. Let U be a generating subset of g, and let S(il) be the semigroup of elements reachable from the identity. If H is irrelevant for transitivity, then (from the previous section) the control system (BGCS) determined by Q, on G is controllable <3> S(fi) = G <3> S(fl).x = M for each x 6 M -O the basic control system (BMCS) on M determined by fl is controllable. Thus we have the following Proposition 3.5. Let H be a closed subgroup of a connected Lie group G that is irrelevant for transitivity. If Q be a generating subset of g, then the basic control system (BGCS) on G is controllable if and only if the basic control system (BMCS) on M = G/H is controllable. The preceding proposition provides good motivation for identifying (IFT) subgroups, since they allow the study of controllability questions on the lower dimensional manifold G/H.

59 The following criterion turns out to be useful so we state it for future reference: Proposition 3.6. Let H C G be a subgroup and S C G a subsemigroup. Then S is transitive on G/H

-v=>

SH = HS = G,

and

H is (IFT) -£=> for every proper S with nonvoid interior SH n HS ^ G. Conversely, H is (NIFT) <£=> there exists a proper S C G with 'mt(S) ^ 0 such that SH = HS = G. Proof. One sees directly that if S is transitive on G/H, then SH = G. If sgH = H, then g £ gH = s _ 1 ff C S^H, so S^H = G. Taking inverses of both sides gives HS = G. Conversely suppose that HS = SH = G. Then for cosets g\H and <72#, pick S\,s? 6 S and hi,hi € H such that g^1 = /i2s2 and gi = sihi. One computes directly that (siS2)(
Exploiting compactness and irrelevancy

Having equivalent statements on different levels can be exploited in two ways: if one wants to prove that a system on a big group G is controllable, one might hope that it is easier to show this for the induced system on the homogeneous space M = G/H. Conversely, noncontrollability "downstairs" follows from noncontrollability "upstairs." Having the possibility to switch back and forth has the simple advantage that it is not always clear in advance which framework is more suitable for the problem under consideration. In particular results that were obtained on one level become suddenly available on the other level, too. As an example highlighting these advantages we reconsider the proof given in 15 of non-controllability of Dubins' problem in hyperbolic space HP when the curvature bound does not allow overcompensation for the intrinsic (constant, negative) curvature of the underlying space. Dubins' problem can be stated as follows: we consider an n-dimensional manifold M of constant curvature e 6 {-1,0,1}. So for e = 0 or 1 we have euclidean space E" or § n whereas e = — 1 leads to hyperbolic space HP. Our task now is the following: Given initial and terminal points XQ,XI € M as well as initial and terminal tangent vectors vt £ TXiM, i = 1,2, find a length-minimizing C 1 curve 7: [0,L] -> M matching these boundary conditions (or Dubins' data) such that 7 has curvature (almost everywhere) bounded by a fixed constant KQ > 0, see FIGURE 1.

60

Y

<

M

XQ Figure 1. Dubins' problem

There are many nice interpretations for this problem: one may think of a car driving in the plane at constant speed that cannot make arbitrarily sharp turns; laying railroad tracks or a plumber laying pipes are other possibilities. The spherical problem one can interpret as a path planning problem for planes or ships. This variational problem can be formulated as an optimal control problem on a homogeneous space of a Lie group, the motion group G™ of M™. For the classical geometries we have: Rn S S E ( n ) / S O ( n ) ,

S" = SO(n + l ) / S O ( n ) ,

and

H" = S O o ( l , n ) / S O ( n ) .

The associated control system on G™ comes from the Serret-Frenet-equations for moving frames. In matrix notation one obtains the system: ' 0 1 0 -e 0 uT 0 -u X

ue

||U||2
Xeso(n-l).

(4.1)

So g g G? C GL(n + 1,R), and M£" = G?/K with K s SO(n). In the euclidean as well as the spherical case one immediately obtains that admissible paths for Dubins' problem always exist because the reachable semigroup has interior points and neither SE(n) nor SO(n) has proper subsemigroups with nonempty interior. This changes in hyperbolic space. Monroy observed that Dubins' problem in the hyperbolic plane is not controllable if the curvature bound «o is too small, namely if KQ < 1. This means that there are Dubins' data xo, £1, "o, «i for which there does not even exist an admissible arc. His argument however, construction of a bounded region B: H 2 —> R invariant under the flow of the control system, did not generalize immediately to higher dimensions. Nevertheless, we have the T h e o r e m 4 . 1 . Dubins' problem in HP is non-controllable iff KO < 1. Geometrically this is a nice theorem because it says that we must be able to overcompensate the curvature of the underlying manifold (constant —1). The proof

61 in 15 exploits irrelevancy and results from Lie semigroup theory. Lawson's investigations of the Mobius semigroups—certain geometrically interesting subsemigroups of 0 ( 1 , n)—and his results on their tangent wedges just happened to provide the answer modulo irrelevancy: Proposition 4.2. Let G = 0 ( l , n ) , fl = L(G) = s o ( l , n ) , and consider the wedge W = t) + C Q & with the cone C = i I r 0 v1 I | r 6 R, v 6 R n _ 1 , r >

\\v\\ 1

and the subalgebra

«6R"_I, X e s o ( n - l ) > = so(l,n-l). Then S{W) is a Mobius semigroup. In particular, S{W) C G is proper with nonempty interior. The important information, of course, is that S(WO ^ G because it immediately implies that S(C) / G. Since if «o < 1, C contains the admissible controls for the system (4.1) on the connected group SOo(l,n), the system is non-controllable. And since the compact group SO(n) C 0 ( l , n ) is irrelevant for control (Corollary 3.4), we immediately obtain the 'if'-part of Theorem 4.1. The 'only-if'-part also follows from a "compactness" argument, but a much simpler one. For v £ R n _ 1 consider Yv Then Y„3 = (1 - |M| 2 )y„. So if ||v|| > 1, then exp(RY„) is a circle group, hence exp(K + V„) = exp(Ry„). Since {{{Yv : \\v\\ = «o})) = 0, this immediately implies controllability of the system (4.1).

5

Irrelevant groups and algebras

We extend the notion of irrelevant for transitivity to an analogous control-theoretic notion, abbreviated (IFC) and also extend the notions to Lie subalgebras. Definition 5.1. A subgroup H of G is irrelevant for controllabilty (IFC) if for all flCg, 5(fi) acts transitively on G/H implies S(Q) = G. A subalgebra h < fl is (IFT) if (exp h) C G is (IFT). Similarly, f) < fl is (IFC) if (exp t)) C G is (IFC). If a group or subalgebra is not (IFT), we say it is (NIFT). L e m m a 5.2. (i) A closed subgroup H of a Lie group G is irrelevant for controllability if and only if for every SJ C g the following statements are equivalent: (a) The system (BMCS) p{t) = U(t) (p(t)), p(t) e M = G/H, U(t) G Q is controllable on M.

62 (ii) The system (BGCS) g(t) = U(t) (g(t)), g(t) 6 G, U(t) € Q is controllable on G. (b) A subgroup H is (IFC) if and only if for all fi C g, S(£l)H = HS(il) S(Q) = G.

= G =>

Proof. We know from Proposition 2.3 that the attainable set at x € M = G/H is 5(fi).x. Thus the system (BMCS) is controllable if and only if S(Q).x = M for all x 6 M and the system (BGCS) is controllable if and only if S(Q) = G. Part (a) is now immediate. The second part follows from Proposition 3.6. • We observe that (IFC) and (IFT) are related, but distinct. One difference is that for (IFT) we only look at semigroups S with int(S) ^ 0. This means in the case of systems Q C g that we only look at systems satisfying the local accesssibility property: int(S.p) ^ 0 for all p e M. For (IFC), on the other hand, we look at arbitrary n C g, so we do not assume in advance that the reachable sets have interior points, i.e., local accessibility. We note that local accessibility of the system induced by Q C g on G/H, H closed, is actually equivalent to ((H)) + t) = g, since the latter is the "full-rank" condition in this context. We will see an example that is (IFT) but not (IFC) in a later section. On the other hand for (IFT) we consider arbitrary semigroups with interior, not just those that arise as the accessibility set from the identity for il C g. Although we do not know whether (IFC) implies (IFT) in general for closed subgroups H, this is the case for a large class of Lie groups. Proposition 5.3. Let G be a connected finite-dimensional real Lie group such that G / R a d G is compact, where R a d G denotes the radical ofG. Then a subgroup resp. subalgebra is (IFT) if it is (IFC). Proof. Let S be a subsemigroup with non-empty interior. By standard maximality arguments (see 7 ) S is contained in a maximal subsemigroup T ^ G, which again must have interior, and which must also be closed. Furthermore, T is generated by exp(L(T)), where L(T) is a closed half-space in g, namely the tangent wedge of T (see Theorem 12.2 and the first part of the proof of Theorem 12.5 in 7 ) . Thus T = 5(fi) for Q := L(T) (see sections 4 and 6 of " ) . Now suppose that the subgroup H is (IFC) and suppose that there existed S with non-empty interior transitive on the set of cosets G/H, but S ^ G. Then the maximal semigroup T containing S is also transitive on G/H (since S C. T). But then by (IFC), T = S(fl) = G, a contradition. Since the proposition holds for all subgroups, it also holds for subalgebras. • Note that this proposition applies to solvable and nilpotent Lie groups since G = Rad(G) in this case. We consider now in more detail the (IFC) property. L e m m a 5.4. Let G be a connected Lie group, let Q C g, and suppose that the (BGCS) determined by il is not controllable. Then Q. is contained in a maximal

63 non-controllable system A and either A is a proper subalgebra of g or A is a closed Lie wedge (i.e., a closed convex cone that is the tangent cone of a closed semigroup) that generates g and hence 5(A) has interior. Proof. Order all subsets of g by inclusion and consider a maximal chain ( = totally ordered set) C of subsets each of which is non-controllable (such maximal chains exist by the Hausdorff Maximality Principle, a variant of Zorn's Lemma). Then their union either generates g or it doesn't. If it does not, then it generates a proper subalgebra o. Then the orbits for the system A = o are the cosets of the analytic group (exp a), and hence the system is not controllable. By maximality of the chain, o belongs to the chain and is its largest element. Again by maximality of the chain, any larger subset generates all of G. Suppose that A, the union of the chain, generates g. Then since g if finite dimensional, there are finitely many elements of A that generate g. Since A is the union of the chain, there must be some element Q in the chain that contains these finitely many elements, and hence fi generates g. Thus S(Q) has interior; let V ^ 0 be an open set contained in 5(fi). Suppose that T is another member of the chain such that T contains fi. Then S(T) misses V-1, since otherwise e € v~1V C 5 ( r ) for w _ 1 £ V-1 (~l S ( r ) . Since G is connected, the open set v_1V generates (as a semigroup) all of G, but this contradicts the fact that 5 ( r ) is proper. One verifies directly that

5Pc(A) = (J{s p c (n): n is in the maximal chain C}, and thus 5 pc (A) also misses V~l. Hence its closure also misses V-1, and hence 5(A) misses V-1 (Theorem 2.2). Thus we conclude that 5(A) is proper, and hence A in not controllable. Again it follows from the maximality of the chain C that A is in the chain and is a maximal non-controllable subset of g. Let 5 be the semigroup 5 ( A ) - , the closure of 5(A). Then it is standard from the Lie theory of semigroups C, 2 ) that the tangent wedge of 5 , L(5) = {X 6 g : exp(tX) e 5 for all t > 0} is a closed convex cone, and that it is the largest subset Q of g such that S(il) C 5 (see Theorem 2.2). It follows that A C L(5), and hence A = L(5) by maximality. D T h e o r e m 5.5. Let G be a connected Lie group with Lie algebra g. Then g contains a largest (lFC)-ideal, denoted IFC(g), i.e., IFC(g) is (IFC), and every other ideal i that is (IFC) is contained in IFC(g). Proof. i,From the preceding lemma we know that maximal non-controllable subsets of g exist, and that each such is either a subalgebra or a tangent wedge L(5) of a closed semigroup 5. Since X —> —X is an anti-isomorphism on g corresponding to inversion on G, it follows that — L(5) is also a maximal non-controllable subset. Thus in the intersection of all maximal non-controllable sets we can replace L(5) and — L(5) in the intersection by the single set L(5) l~l — L(5), a subalgebra (Theorem 2.2). We conclude that the intersection of all maximal non-controllable subsets of g is a subalgebra q, the largest subspace contained in all of the maximal non-controllable

64 sets. Any automorphism Ad(g), g e G, of g must carry a maximal non-controllable subset to a maximal non-controllable subset. Hence their intersection q is Ad(G)invariant; it follows that q is an ideal in g. Suppose that 5(fi)Q = QS(Q) = G for ft C g, where Q = (expq). We want to show that 5(ft) = G and apply Lemma 5.2(b) to show that q is (IFC). Suppose that 5(ft) ^ G; then ft is not controllable and hence by Lemma 5.4 is contained in a maximal non-controllable subset A. But by definition q C A, and thus G = 5(ft)Q n QS(Q) C 5(A)Q n Q5(A) C S(A) # G, a contradition. Thus q is (IFC). Suppose that i is an ideal that is not contained in q. Then there must be some maximal non-controllable subset A of g such that t is not contained in A. Since i is an ideal, the group I := (exp i) is normal, and hence 5 ( A ) / = 7 5 ( A ) / is a subsemgroup. It clearly contains exp(i)Uexp(A) and hence the semigroup 5 p c (iUA). By the maximality of A, the system i U A is controllable. It follows from part (i) of Theorem 2.2 that iUA generates g, and hence that 5 pc (iUA) is a dense subsemigroup of 5(i U A) = G with non-empty interior, and hence all of G. But again by Theorem 2.2, 5 p c (t U A) is the semigroup generated by exp(i U A), and hence is contained in 5(A)/. It follows that G = 5 ( A ) / = / 5 ( A ) , but 5(A) # G. Thus the ideal i is not (IFC). Thus the ideal q satisfies all the assertions of the theorem. • The following proposition if frequently useful for reductions in working with the property (IFC). Proposition 5.6. Let G be a connected Lie group such that the subgroup K := {exp IFC) (g) is closed (this will automatically be the case if G is simply connected). Then the group H = G/K has a trivial (lFC)-ideal. Proof. Note that K is a normal subgroup since IFC(g) is an ideal. Define a : G —> H by a(g) = gK. PicK F # 0 in h = g/IFC(g), and pick X € g such that da(X) = Y. Since the kernel of da is IFC(g), we have X is not in IFC(g). Thus there exists a maximal non-controllable set A in g missing X. Then IFC(g) C A, so if C 5(A). Since 5(A) is proper in G and contains K, it follows that its image is proper in H. Since a(expA) = exp(da(A)), we conclude that a (5(A)) = 5 ( d a ( A ) ) . Hence T := da(A) is not controllable. Since A contains IFC(g) and X 0 A, it follows that Y £ T. Furthermore, F is maximal non-controllable, since any set that properly contains T will have inverse properly containing A, hence be controllable on G and hence its image, the original set, will be controllable on H. Thus Y & T implies Y £ IFC(h). It follows that IFC(h) is trivial. • 6

Irrelevant groups and algebras: t h e solvable case

For solvable Lie algebras we are in the fortunate situation of having a classification of the maximal subsemigroups with non-empty interior, which makes possible a detailed treatment of irrelevant subgroups. Given a Lie algebra g the natural setting for treating controllability and transitivity questions is to work in the corresponding simply connected group G because

65 every other (connected) Lie group G with Lie algebra g is a quotient of the simply connected group: General A s s u m p t i o n . Throughout this section we assume that G is a simply connected solvable group. In the sequel we will discuss the notions (IFT) and (IFC) separately. We will see that every solvable Lie algebra g contains two special ideals, a maximal (IFT)ideal as well as a maximal (IFC)-ideal (already introduced). The first one is also maximal (IFT) as a subalgebra. Reduction modulo these ideals leads to Lie algebras having a very simple structure. On this level one can find easily the answers to many interesting questions. For example, it is easy to figure out the minimum number of elements necessary to generate g as a Lie algebra. 6.1

Discussion

of

(IFT)

For solvable groups virtually all questions we consider can be answered satisfactorily because we have a complete classification of their maximal subsemigroups with interior points 7 . Definition 6.1. Let g be a Lie algebra. A subalgebra f) of codimension 1 is called a hyperplane subalgebra. A halfspace bounded by such a hyperplane subalgebra is called a halfspace semialgebra. T h e o r e m 6.2. Let a be a solvable Lie algebra and G an associated connected group. (i) If S C G is a maximal subsemigroup with int(S) ^ 0, then L(5) = {X £ g | exp(R + X) C S } is a halfspace semialgebra and S is the semigroup generated by exp(L(5)). Every semigroup with non-empty interior is contained in some such maximal semigroup. (ii) Conversely, if G is simply connected and f)+ C g is a halfspace semialgebra, then S(f) + ) = (expl) + ) C G is a maximal subsemigroup. Thus in the simply connected case, the maximal subsemigroups of G with nonempty interior are in one-to-one correspondence with the halfspace semialgebras in g. Hyperplane subalgebras are well-understood and fully classified, cf. 3 . Let us write 7i(g) for the set of all hyperplane subalgebras of g. Then the so-called Aradical

A(g) = f)^(fl) plays a crucial role. In fact A(g) is an ideal, and due to Theorem 6.2 we have: Corollary 6.3. The ideal A(g) is irrelevant for transitivity (IFT). Proof. Indeed, let A(G) C G denote the corresponding connected normal subgroup. If S C G is a proper subsemigroup with nonempty interior, then S is contained in a maximal subsemigroup S(f) + )- As A(g) C f)+ we obtain S A(G) C S(f) + ) / GThus S is not transitive on G/A(G). D As we already mentioned, A(g) need not be (IFC), in general.

66 Example 6.4. The smallest dimensional example for which A(g) is not (IFC) is the algebra of euclidean motions of the plane: g

= sp&n{A,X,Y},

[A,X] = Y,

[A,Y] =

-X,

and all other brackets vanishing. Here A(g) = g' = span{X, Y}. If VI — RA, then the induced system on G/A(G) ~ R is controllable but the system g = ug, u G RA, is not controllable. Since A(g) is an ideal, g/A(g) is a Lie algebra (in 3 such Lie algebras are called A-reduced), and the homogeneous space G/A(G) is actually a Lie group. Our next goal is to show that there are no (IFT)-subalgebras beyond A(g). This will be carried out in several steps. The basic idea is to be found in the proof of the lemma below. L e m m a 6.5. Ift) £ g' then H := (expf)) is (NIFT). Thus ift) is (IFT), then t) C g'. Proof. Pick X G I) \ g'. We want to find a semigroup S ^= G such that int(S) ^ 0,

and

5exp(RX) = exp(RX)S = G

in order to prove that RX and hence H is (NIFT) (see Proposition 3.6). So let us pick a hyperplane t) D g' such that f) + RX = g and consider the halfspace semialgebra t)+ = t) +R+X, the semigroup S(f) + ) = (exp(fj + )\, and the subgroup H = (expljY Then fj is an ideal because of g' C fj, so H is a normal subgroup. Hence e x p ( R X ) ^ = .ffexp(lIL\r) = G, whence exp(RX) S(fj) = S(I))exp(RX) = G. So exp(RX) is (NIFT) which implies that H is (NIFT). • It is crucial in the previous proof that t) is an ideal. It is not difficult to find subgroups Hi, H2 of a group G such that f)i + ()2 = g but H\Hi ^ G. E x a m p l e 6.6. Let L be the two-dimensional non-abelian Lie group:

L=

{(oi)|a>0'

beR

\-

We may identify L with the upper halfplane: L = {(b, a) | a > 0} C R2. This allows to describe all interesting objects like one-parameter groups, cosets, etc. nicely in purely geometric terms. So we don't need do write down lengthy computations which are elementary but not very elucidating. The group identity is e = (0,1), the commutator group L' is the horizontal line R x {1} whereas an arbitrary one-parameter group is just an ordinary line—better said its part in the upper halfplane—through e. For g G L and a one-dimensional subgroup H the left coset gH is simply the line through g parallel to H. If H ^ L', then the line representing H intersects the horizontal axis at some point (v,0). The right coset Hg is nothing but the line through g and (v,Q), cf. the FIGURE 2. If we take two subgroups Hi ^ H' ^ H2 then H1H2 ^ L will hold, a fact one can best "see" in the picture FIGURE 3.

67

G

y/gH \ / /

/

/ X X

VV

^

Pi

Figure 2. The right coset Hg

Figure 3. HiH2 # L

Eventually we will use an argument of the above type, but it requires some refinement. L e m m a 8.7. Let_M = G/H, w: G -* M the canonical map, g 6 G, p = ir(g) € M and X € g. Let X denote the vector field induced by X on M. Then X(p) = 0 iff X 6 Ad( fl )h. Proof. Let po = w(e) denote the base point in M. Then its stabilizer algebra is gj 0 = h. For p = n(g) = g.po the stabilizer group is gp = Ad(g)QP0 = Ad(g)h. As X(p) = 0 iff X 6 flP, our claim follows. D

68 Proposition 6.8. Assume that g = hi + 1)2 with subalgebras fh>f)2 such that Ad(G)f)2 n hi = {0}. Then G = HXH2 with Hi = (exp) h<, i = 1,2. Proof. Let M = G/H2.

Forp € M let D{p) - | x ( p ) \ X € I n } . Then D(p) = TPM

for all p € M follows from the previous lemma because the map X i-» X ( p ) : f)i —> T P M has kernel C Ad(G)h 2 D hi = 0 and dim hi = dimT P M. Thus Hi.p is a neighborhood of p for every p 6 M. As M is connected, M = H\.n(e) follows, i.e., HiH2 = G. • Proposition 6.9. Let X € g \ A(g). Then there exists a hyperplane subalgebra t) such that Ad(G)X n f) = 0. So the subalgebra EX is (NIFT). Proof. First we pass to the quotient g/A(g), so we assume w.l.o.g. that Q is Areduced. This implies, cf. [KHH], that g is a so-called basic metabehan Lie algebra which means it has the following structure:

0=oe®8\

(t)

where o is a vector space, A is a set of nonzero linear forms from o —> R, the a A 's are vector spaces, too, and [A, Xx] = \(A) Xx

for A e 0, Xx 6 £|A.

All other brackets vanish: [flA,flM] = 0 for all A, fi. Moreover g' = © A e A f l A . Our goal now is to show that for every nonzero X in a basic metabehan $j we find a hyperplane subalgebra t) such that f) + MX — g and Ad(G)X n f) = 0. But this is a relatively easy task. For X £ g' this has already been proved in Lemma 6.5. For X 6 g' \ {0} we decompose X as X = ^2X X\ and pick a /i such that XM ^ 0. Next we fix a hyperplane V^ C gM with X^ 0 VM and let h= o + ^ g

A

+

V

ThenAd(G)Xnf) = 0 because of Ad(G)X C (0, oo)X„+g' and RX M nV M = {0}.

•

Thus we can conclude our discussion of (IFT)-subalgebras with the following theorem, which typically allows us to reduce transitivity questions on G to G/A(G). T h e o r e m 6.10. If a subalgebra t) C g is (IFT), then t) C A(g). In particular A(g) is maximal (IFT) not only among ideals but also among subalgebras. One can give an alternate more direct proof based on results of 7 . Suppose that f)i is (IFT), but is not a subset of A(g). Then there exists a subalgebra f) of codimension 1 such that hi \ f) / 0. Pick X such that X 6 f)i \ f). Note that g = h + RX. By Lemma 6.5 we must have that l)i C g', and hence X 6 g'. Set h + = f) + RX; by Theorem 6.2 S(l} + ) = (exp l) + ) is a maximal semigroup of G.

69 By results of Section 11 of 7 , there exists a surjective homomorphism with connected kernel (f> : G —f R or ^ : G -> L, the group of Example 6.6 such that 0(S(h+)) = R+ or, respectively, L+ := {(b,a) € L : b > 0}. The first case is impossible, since (d0) _1 (O) is an ideal of a containing the derived algebra, hence X, and containing I) since exp(h) must map to a subgroup of R + , hence be 0. Thus $j, and hence G, maps to 0, contradicting surjectivity. Thus there exists : G —> L such that ( s ( h + ) ) = L+. The subgroup H = (exph) must map into L\ := {(b, a) £ L : b = 0}, the largest subgroup of L+ and exp(X) must map to L' := exp(£') = {(6, a) £ L : a = 1}, the image of the derived algebra (! of the Lie algebra I of L. Suppose that the image of H is trivial. Then the image of f) is trivial, the image of g under d(j> is contained in the image of RX, which is most one-dimensional, and hence the image of G must be one-dimensional, contradicting surjectivity. In a similar way one argues that (expMX) is not the identity. Thus (f>(exp MX) — L' and <j) o exp = exp^ od<j> restricted to MX must be injective, since e x p t is and d(\> restricted to UX must be. Finally note the the kernel K of is a closed connected subgroup, hence a Lie subgroup with some Lie algebra t. If there existed Y € t \ h, then (expRY) = e implies d<j>{Y) = 0, and since R F + fj = fl, it follows that d(o) = d0(h) is one-dimensional, again leading to a contradiction of surjectivity. Thus EC ^ and hence K = (expt) C (exp h) = H. One verifies directly that L = L\L' = L'L\ and that the factorization is unique in each case. Let g 6 G. Then (exp tX)<j)(h) 6 L'L\ for some t 6 R, h £ H. Thus g = (exptX)hk for some k € K C H; thus g £ exp(RX)H. Suppose that g = exp(tiX)hi = exp(t2X)h,2. Then <j>(g) = {exptiX)4>(h{) = (expt2X)4>(h,2)By unique factorization in L, exp{d(p(tiX))

= 4>(exptiX) = 0(expt 2 X) = exp (d(j>(t2X)).

and by injectivity of expod<^> on MX, we conclude t\ = ti- Then by cancellation h\ =b,2, so the representation is unique. The same argument works for .H'exp(RX'). Thus by Proposition 3.6 f) is not (IFT). A reason for presenting the alternate proof is that this second proof gives us the following new characterization of maximal semigroups. T h e o r e m 6.11. Let G be a simply connected solvable Lie group and let S := S(h + ) = ( e x p h + ) be a maximal semigroup of G, where t)+ is a half-space semialgebra, and f) is a codimension one subalgebra which is not an ideal. Then there exists X £ (h + \ h) f~l g' and for any such X, S = H(expRX) = (expMX)H with unique factorization in each case, where H := (exp h) We remark that the proof remains valid in the more general connected case. 6.2

Considering

(IFC)

The notion of irrelevance for controllability (IFC) is the natural one to look at if one's starting point is control systems on manifolds rather than the semigroups of smooth invertible maps. So we start with the vector fields generating flow maps

70 rather than the flow maps themselves. Actually this was our original motivation to introduce the irrelevancy notion, which proved more broadly powerful. For detailed proofs we refer to 17 . As a consequence of Theorem 5.5 and Proposition 5.6, controllability questions on g can be reduced to a problem on the factor algebra JJ/ IFC(g). Of course whether this information is very useful or not depends on the following two aspects: (D) Determination: How hard is the problem of determining IFC(g)? (S) Simplification: How complicated is the structure of the quotient g/IFC(g)? Is it much simpler that that of a general solvable g? For nilpotent g we obtain the best possible answers one can hope for: T h e o r e m 6.12. If g is nilpotent, then IFC(g) = A(g) = g'. A subalgebra f) C g is QFC) iff t) C g,'. Proof. An (IFT)-subalgebra f) is contained in g' by Lemma 6.5, and thus A(g) C g' by Corollary 6.3. By Proposition 5.3 we conclude that every (IFC)-subalgebra is contained in g', in particular IFC(g) C g'. The converse containments in the first statement are proved in 17 , and the converse implication in the second statement then follows from Theorem 5.5. • For g solvable, not nilpotent, the answer to (D) is, roughly speaking, that it is as hard as finding the Jordan normal form of a linear map. And concerning (S) one finds that g/ IFC(g) is only slightly more complicated than the basic-metabelian algebras mentioned in the previous section. The structure of these algebras (in terms of brackets) is not very complicated and actually easily understood, the major task one has to perform is introducing a well-suited notation. One might be disappointed by our anwswer to (D), in particular after one finds out that it is also necessary to pick a "generic" element in g, so this is not a straightforward procedure for finding IFC(g). But actually finding IFC(g) itself is not necessarily the task that must be solved. There are smaller ideals which are easily determined and which already provide most of the possible reduction: Proposition 6.13. If Q is solvable, then D2Q = [g',g'] is an (IFC)-ideal. See 17 for a proof. The factor algebra g = Q/D2Q is metabelian, i.e., its bracket algebra [g, g] is abelian. For practical purposes Proposition 5.6 can be quite useful. It allows one to proceed inductively: having found some (IFC)-ideal, pass to the quotient and look for (IFC)-ideals there. Finally there is a nice characterization of (IFC)-ideals in algebraic terms. Definition 6.14. We call a subalgebra f) C g an (NCA)-subalgebTa if it has the following property: If a C Q is a subalgebra and a + I) = g, then a = g. The acronym (NCA) stands for no complementary algebra: f) being an (NCA)subalgebra means there is no proper subalgebra o with 0 + f) = g. Of course it makes sense to speak of (NCA)-ideals, too. Proposition 6.15. An ideal i o/g is (IFC) if and only if it is an (NCA)-ideal.

71 We cannot expect that a sum of (IFC)-algebras fji,h2 is (IFC). Nevertheless, we obtain an analogue of the previous proposition. Its proof requires a few extra arguments, though. Proposition 6.16. If the subalgebra I J C J generates a closed subgroup (exp f)) and is (NCA), then it is (IFC). Every (NCA)-algebra t) is contained in A(g). Proof. Suppose that t) is (NCA). Then h C A(fl) must hold because if I) \ A(g) ^ 0 we can actually find a hyperplane subalgebra f) such that i) + t) = 0, cf. Lemma 6.5. We claim that f) is (IFC), so we consider M = G/H and $ C g such that the system on M given by {x | x e $} is controllable. We must show that S ( $ ) = G holds true. As usual 7r: G —> M denotes the canonical map, pa = 7r(e) the base point, and pr,, = d7r(e): a —> fl/f) is the projection with kernel I). Now let a = (($)) C 0, the subalgebra generated by $ . Then controllability on M implies TP0M = < X(po) | X 6 o !• which is equivalent to pr1)(o) = 0, hence a+ f) = 0. As f) is (NCA), o = £| follows, so int(5(*)) # 0. Since f) C A(g), H C A(G). Thus 5 ( $ ) is transitive on G/A(G). As A(G) is (IFT) and int(S($)) # 0, we deduce S ( $ ) = G, finishing the proof. • Now we ask if the reverse implication holds true, so is every (IFC)-algebra (NCA)? Example 6.6 and the discussion leading to Proposition 6.8 show that this is a nontrivial problem. Take an t) which is not (NCA). Take a proper subalgebra o 7^ g such that a + t) = Q. Then it is clear that the system induced by $ = a on M = G/H will be transitive at least on a neighborhood of the base point po = 7r(e) of M, but we cannot expect in general that the semigroup S = A = (exp o) is transitive on M, for that is equivalent to G = HA, and the latter we know for sure only if Ad(G)fj n o = {0} is trivial. Example 6.17. Again we consider the algebra of motions of the euclidean plane 0 = s p a n { A , X , Y } with brackets: [A,X] = Y, [A,Y] = -X. Which subalgebras f) of fl are not (NCA)? Let us look at dim h: 3-dim: Of course 0 is not (NCA), but it isn't (IFC) either. 2-dim: There's only one hyperplane subalgebra: 0'. Of course 0' is not (NCA), but it is not (IFC) either. 1-dim: The line RX is not (NCA) iff X £ g' because 0' is the only possible choice we have for a complementary subalgebra. As Ad(G)0 ; = 0', RX n Ad(G)$' = {0}, so G = exp(RX)G' holds true, i.e., RX is not (IFC). So in this case -i(NCA) implies -i(IFC). In particular we observe that, for example, the subalgebra f) = RX is (NCA), so it is (IFC), too. But we also observe that f) g IFC(0) = {0}. So the inclusion f) C A(o) if t) is (IFC) cannot be improved any further. An (IFC)-algebra need not be contained in the ideal IFC(0). We conjecture that-.(NCA) =$• -.(IFC), resp., (IFC) =$> (NCA) so we state this as an open

72 Problem 6.18. Prove or disprove: an (IFC)-algebra is also (NCA). Now we return to our discussions of the benefits of passing to the quotient g/IFC(g). 6.3

The controllability

normal

form

Reduction modulo A(g) led us to A-reduced algebras which actually turned out to be the ones we called basic-metabelian. The algebra g/IFC(g) also deserves a special name, we call it the controllability normal form (CNF)o/g. If IFC(g) = {0} we say that g is in (CNF). Although the structure (bracket-wise) of such a g in (CNF) is not really complicated, having a reasonable terminology is indispensable for analyzing it. Roughly speaking, such a g is made out of two components, a real vector space V and a set (actually a vector subspace) o C g£(V) of linear maps V -*V such that (1) every A G a is diagonalizable over C, and (2) any two maps A,A'(Ea

commute: A A' = A' A.

These conditions are tantamount to saying that a is an abelian subalgebra of g£(V), and V may be decomposed into common eigenspaces of A, A'. Formalizing this statement it turns out that one can decompose g in the following way:

fl = h e 0 g A © ® g u , AeA

(6.1)

wen

which has to be interpreted as follows: • h is an abelian subalgebra of g corresponding to o plus the center 3(g). • The sets A, Q stand for the so-called roots signifying the common eigenvalues and g , g" are common eigenspaces also called root-spaces. • A is the set of real roots; each A G A is a nonzero linear form A: f) —> R, and [H,XX] = \{H)XX,

holds true for all H G h, Xx G g A .

• Q is the set of complex roots—actually only one half of it since complex eigenvalues always appear in conjugate pairs—so an w G fi is an R-linear map from h -> C such that w(h) % R. The root space g" is isomorphic (as a vector space) to some Cr considered as a real vector space, and for H € h, Xu G gw

[H,Xu]=u(H)Xu. • Each root space gA, g" is an abelian ideal and [g a ,g^] = 0 for all a,/3 G A u Q . At this point one clearly sees the difference between the (CNF) g/IFC(g) and the basic metabelian g/A(g): in the latter case there are no complex roots (eigenvalues) .

73 Proposition 6.19. Let g be as in (CNF). Then

fl' = ©8 A ©©8 W > A€A

"* A(g) = © g " .

wgn

wen

A hyperplane subalgebra a in g has the form 0- = cii, + 8 ,

wi£/i a hyperplane Of, C h, or

a = f) © ^ f t g a © a,,

6.^

Generic controllability

un£/a a hyperplane oM C g^.

rank and type

Now we reap some of the benefits of reduction because we will be able to answer the following questions: • What is the minimum number of elements Xi,... ,Xr necessary to generate a given g as a Lie algebra, i.e., {{X\,... ,Xr)) = 8? • Given a Lie algebra g, some "drift vector fields" Ai,... ,Aj and some "input vector fields" Bi,... ,Bi, what are the minimal numbers (i,d) such that U = {Ai ,...,Ad, ±B%,..., ±Bi) is controllable? • Will controllability or noncontrollability of the system given by U = {Ai ,...,Ad, ± B i , . . . , ±Bi} C jj be stable under perturbations (to Aj, Bk € fl)? Is it a generic property? For the first two questions we observe that existence of such minimal numbers r, resp (i,d) is not really an issue, but it is not at all clear how to compute them. We call r the generic controllability rank of g, GCR(g), and (i,d) the generic controllability type of g, GCT(g). This choice of terminology has something to do with the last question because once certain conditions can be expressed in terms of some polynomials (say in Xi,..., Xr) not vanishing (like a rank condtion), they are either void or generic, so they hold either never or in almost every case. Looking at the (CNF) of a solvable g we can immediately read off its controllability rank and type. With the notation of (CNF) we let d\ = max < diniR g | A 6 A > ,

and

du = max {dime g" | u; € fi} ,

and obtain: Proposition 6.20. Let g be in (CNF).

Then

GCR(g) = max {dim t), 1 + dA, 1 + dn} , and GCT(g) = (i, d) with i + d = GCR(g), i = max {dim f), 1 + ^ A } ,

and

d = max{0,1 + dn — i) •

74 The proof requires two separate arguments. One must find a set of generators of the appropriate size, and one must show that one cannot generate g with less elements. In both cases the root decomposition as in (6.1) is instrumental. The details can be found in 1T. A particularly nice result is the next theorem which simplifies Sachkov's work on controllability of solvable Lie algebras, cf. 2 2 , also 2 3 . Definition 6.21. We call a Lie algebra g SID-controllable, if there exist A,B € g such that S(A,±B) = G, i.e., the single input plus drift system g = g(A + uB), u 6 R is controllable. It turns out that this is a generic property of the Lie algebra g rather than the particular pair (A,B): the set of such pairs is either empty or open and dense in g x g. T h e o r e m 6.22. For a solvable Lie algebra the following statements are equivalent: (i) g is SID-controllable. (ii) every factor algebra g/i (i C g an ideal) is SID-controllable. (hi) g/£>2g is SID-controllable (iv) GCT( 0 ) = (1,1), (v) g/IFC(g) is isomorphic to a semidirect product Ck x$j R with Q diag(a>i,...,uik) 6 g£{k, C) and w i , . . . ,Wfc € C \ R pairwise distinct.

=

If g is SID-controllable, then the foloowing statements hold true: (vi) $j has no ideals of codimension 2. (vii) Q' is a hyperplane and A(g) = $)'. Alternatively, in view of (iii), one can reduce the problem of classifying SIDcontrollable algebras to the metabelian case [fl',g'] = 0. This is a basic case in Sachkov's work which allows a very short classification where the conditions involved are similar to familiar criteria from linear control theory. Proposition 6.23. A metabelian g is SID-controllable iff it is isomorphic to a semidirect product M.k x ^ R where A € g£(k,M) has nonreal spectrum spec(A) C C\R and every eigenvalue has geometric multiplicity 1. The latter is equivalent to iank(A + XI) > k - 1 for all A 6 C. Comparison with the previous theorem reveals how far away from g/ IFC(g) the possibly bigger algebra Q/D2Q actually may be. The metabelian g/D 2 g has a root decomposition similar to the (CNF):

g/D2g = h © 0 g A e 0 g „ . A€A

u6d

The difference is that the root spaces Q\, gw need not be eigenspaces, they may be generalized eigenspaces; and the subalgebra t) is nilpotent but not necessarily abelian.

75

7

Irrelevant groups and algebras: the semisimple case

The highly developed structure theory for semisimple Lie groups and algebras allows one to derive quite powerful results concerning irrelevancy in the semisimple setting. This project has been carried out by L. San Martin and co-workers over a period of several years. We do not undertake to explain the details of these developments here, but only highlight a few principle results. Let G be a connected semisimple Lie group with finite center and let a denote its Lie algebra. Then the Lie group and the Lie algebra admit Iwasawa decompositions of the form G = KAN, g = t + a + n, where K = exp(t), A = exp(a), and N = exp(n), and where the product factorization of each element is unique and the Lie algebra decomposition is into direct summands. The Iwasawa decompositions arise by taking an Cartan decomposition t + p of 5, picking a maximal abelian subalgebra o of p, letting II be a set of positive roots for a, and setting n equal to the sum of the root spaces for the positive roots. Set M equal to the centralizer of A in K. Then P = MAN is a group, called a minimal parabolic group. The coset space G/P is called a maximal boundary of G. The first major result of San Martin appeared in 2 4 . We cite it in the language of this paper, not the language that he used. T h e o r e m 7 . 1 . Let G be a connected semisimple Lie group with finite center, and let P be a minimal parabolic subgroup. Then P is (IFT). Much stronger results are available if G is simple 25 . T h e o r e m 7.2. Let G be a simple Lie group and let H he a closed subgroup such that 0 < dim(H) < dim(G) and the coset space G/H is compact. Then H is (IFT).

References 1. J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, (Oxford Press, Oxford, 1989). 2. J. Hilgert and K.-H. Neeb, Basic Theory of Lie Semigroups and Applications, (Springer Lecture Notes in Mathematics 1552, Springer, 1993). 3. —, "Hyperplane subalgebras in real Lie algebras," Geom. Dedicata 36, 207-224 (1990). 4. K. H. Hofmann, J. D. Lawson, and E. B. Vinberg, Editors, Semigroups in Algebra, Geometry and Analysis, (de Gruyter, Berlin, 1995). 5. V. Jurdjevic, Geometric Control Theory, (Cambridge Press, Cambridge, 1997). 6. V. Jurdjevic and H. Sussmann, Control systems on Lie groups J. Diff. Eq. 12, 313-329 (1972). 7. J. D. Lawson, Maximal subsemigroups of Lie groups that are total, Proceedings of the Edinburgh Math. Soc. 87 , 479-501 (1987). 8. —, "Ordered Manifolds, Invariant Cone Fields, and Semigroups," Forum Math. 1 , 273-308 (1989). 9. —, "Semigroups of Ol'shanskii type," Semigroups in Algebra, Geometry and

76 Analysis, (de Gruyter, Berlin, 1994). 10. —, "Semigroups in Mobius and Lorentzian geometry," Geom. Dedicata 70, 139-180 (1998). 11. —, "Geometric control and Lie semigroup theory," In: Differential Geometry and Control, Proceedings of Symposia in Pure Mathematics, 64, Editors: G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, (Amer. Math. Soc, 207-221 1999). 12. C. Loewner, "On some transformation semigroups," J. Rat. Mech. and Anal. 5, 791-804 (1956). 13. —, "On some transformation semigroups invariant under Euclidean or nonEuclidean isometries," J. Math, and Mech. 8, 393-409(1959) . 14. —, "On semigroups in analysis and geometry," Bulletin A.M.S. 70, 1-15 (1964). 15. D. Mittenhuber, "Dubins' problem inhyperbolic space," In: Geometric Control and Non-holonomic Mechanics, CMS Conference Proceedings, Volume 25, Editors: V. Jurdjevic, R.W. Sharpe, (Can. Math. Soc, 101-114 1998). 16. —, "Controllability of solvable Lie algebras," J. Dyn. Contr. Systems 6, 453459 (2001). 17. —, "Controllability of systems on solvable Lie groups: the generic case," J. Dyn. Contr. Systems 7:1, 61-75 (2001) . 18. F. Monroy-Perez, "Non-euclidean Dubins' problem," J. Dyn. Contr. Systems 4:2, 249-272 (1998). 19. G. I. Ol'shanskii, "Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal structures on pseudo-Riemannian symmetric spaces," Sov. Math. Dokl. 26 , 97-101 (1982). 20. —/'Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series," Fund. Anal, and Appl. 15 , 275-285 (1982). 21. R. S. Palais, "A global formulation of the Lie theory of transformation groups," (Memoirs of the Amer. Math. Soc, Providence, 1957). 22. Y. Sachkov, "Controllability of right-invariant systems on solvable Lie groups," J. Dyn. Contr. Systems 3:4, 531-564 (1997). 23. —, "Survey on Controllability of Invariant Systems on Solvable Lie Groups," In: Differential Geometry and Control, Proceedings of Symposia in Pure Mathematics, Volume 64, Editors: G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, (Amer. Math. Soc, 297-317 1999). 24. L. San Martin, "Invariant control sets on flag manifolds," Math, of Control, Signals, and Systems 6 , 41-61 (1993). 25. L. San Martin and P. Tonelli, "Transitive actions of semigroups in semisimple Lie groups," Semigroup Forum 58, 142-151 (1999). 26. H. Sussmann, "The bang-bang problem for certain control systems in GL(ra, R), SIAM J. Control 10, 470-476 (1972).

77 CANONICAL CONTACT SYSTEMS FOR CURVES: A SURVEY

WITOLD RESPONDEK Laboratoire de mathematiques de VINSA de 16 131 Mont Saint Aignan, France E-mail: [email protected]

Rouen

WILLIAM PASILLAS-LEPINE Laboratoire des signaux et systemes, Supelec 91 129 Gif-sur-Yvette, France E-mail: [email protected] We start with a brief overview of results characterizing distributions (equivalently Pfaffian systems) that are locally equivalent to a canonical contact system. Then we concentrate on distributions equivalent to the canonical contact system on 7 n (IR,]R m ), that is the canonical contact system for curves. We give various necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact systems for curves. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions contained in the successive elements of the derived flag. We give a new characterization of distributions locally equivalent to the canonical contact systems for curves that uses only one involutive subdistribution in which all information about the geometry and about the regularity is encoded. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R, R m ) in a way analogous to that how Kumpera-Ruiz normal form generalizes Goursat normal form, that is, the canonical contact system on Jn(R, M.). Finally, we briefly discuss how the geometry of distributions equivalent to the canonical contact systems for curves is reflected by their flatness and illustrate some of our results by a model of a simple nonholonomic control system.

1

Introduction

Consider Jn(Rk ,Rm), and denote by (qi,...

,qk,ui,...

the space of n-jets of smooth maps from Kfc into E m , ,um,Pi),

for 1 < i < m and for 1 < \a\ < n,

the canonical coordinates, also called natural coordinates, on this space (see e.g. 3 , 25 , and 3 2 ) , where qj, for 1 < j < k, represent independent variables and Ui, for 1 < i < m, represent dependent variables; the vector of non-negative integers a = (<TI, . . . ,o~k) is a multi-index such that \o~\ = o~\ + • • • + o~k
78

Denote p° — ut, for 1 < i < m. Any smooth map ip from Efc into Rm defines a submanifold in Jn(Rk ,Rm) by the relations Pi = - ^ — ( g i , - - - , % ) ,

for 1 < i < m and 0 < \a\ < n. This submanifold is called the n-graph of ip. It turns out that all n-graphs are integral submanifolds, of dimension k, of a distribution called the canonical contact system on Jn (Rk, Rm) or the Cartan distribution 32 on J n (M f c ,E m ). The Pfaffian system that anihilates this distribution, which is also called the canonical contact system (see e.g 3 and 2 5 ) , is given in the canonical coordinates of Jn(Rk, Rm) by ft

dp? - YjPi

.j.

'dqj = 0,

for 1 < i < m and for 0 < |cr| < n — 1,

where a + \j = {a\,... , Oj + 1 , . . . ,ak). The above description explains the importance of contact systems in geometric theory of (partial) differential equations and in differential geometry. In the former, a (partial) differential equation is interpreted as a submanifold in J"(K fc , IRm) and thus it is natural to study the geometry of pairs consisting of a contact system and a submanifold, see e.g. 3 , 19 , and 3 2 . In the latter, contact systems allow to describe, for instance, diffeomorphisms which preserve the n-graphs of appplications (for example, n-graphs of curves in the case k = 1), see e.g. 3 and 2 5 . A natural problem which arises is to characterize those distributions which are (locally) equivalent to a canonical contact system. This problem was posed by Pfaff 29 in 1814 and seems still to be open in its full generality although many important particular solutions have been obtained. In the case n = 1, with m = 1 and an arbitrary k the final solution has been obtained by Darboux 7 in his famous theorem generalizing earlier results of Pfaff 29 and Frobenius n . The case n = 2, m = 1 and A; = 1 was solved by Engel in 8 . The case n > 2, m = 1 and k = 1 was solved by E. von Weber 33 , Cartan 5 and Goursat 14 (at generic points) and by Libermann 18 , Kumpera and Ruiz 17 , and Murray 24 (at an arbitrary point). The case n = 1, with k and m arbitrary has been studied and solved by Bryant 2 (see also 3 ) . This paper is devoted to the problem of when a given distribution is locally equivalent to the canonical contact system in the case k = 1, n and m arbitrary, that is to the canonical contact system for curves. This problem has been studied by Gardner and Shadwick 12 , Murray 24 , and Tilbury and Sastry 31 (as a particular case of the the problem of equivalence to the

79

so-called extended Goursat normal form). Their solutions are based on a result of 12 that assures the equivalence provided that a certain differential form satisfies precise congruence relations. The problem of how to verify the existence of such a form had apparently remained open. This difficulty was solved by Aranda-Bricaire and P o m e t 1 , who proposed an algorithm which determines the existence of such a form. Their solution, although being elegant and checkable, uses the formalism of infinite dimensional manifolds and thus goes away from classical results characterizing contact systems. Recently, the authors 27 gave a geometric characterization of canonical contact systems in the case k = 1, and n and m arbitrary. That characterization of the canonical contact system on Jn(R,Rm), recalled as Theorem 2.1, turns out to be a natural combination of that given for J 1 (E f c ,R m ) by Bryant (see 2 and 3 ) and that given for J"(M,M) by Murray 2 4 . The aim of this paper is to analyze various possible characterizations of the canonical contact system on J"(M, K m ) . We start with stating, as Theorem 2.1, the result of 2 7 . That result involves two conditions: an essential geometric condition stating that each element 2)W of the derived flag (see the definition in the next section) contains an involutive subdistribution of corank one and the regularity condition T>^ = T>i (see the definition of the Lie flag in the next section). We will discuss other equivalent formulations of both conditions using, in particular, the notion of characteristic distributions (Theorem 5.4) and the language of Pfaffian systems (Theorem 3.7 and Corollary 5.5). All those descriptions lead to the following characterization, stated as 5.6, which is the main result of the paper. Main Theorem A rank m + 1 distribution V, with m > 2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of any point p in M, to the canonical contact system on J"(M, M.m) if and only if the following conditions hold. (i) £>(") = TM. (ii) £>(™-1) is of constant rank nm + 1 and contains an involutive subdistribution Cn-i that has constant corank one in (iii) 2>(°>(p) g £„_!(?). This result implies that all information containing both the geometry and the regularity of the problem is completely encoded in the involutive subdistribution £„_i of corank one in D(™-1). In particular, its existence and the regular intersection with 2?(°) (p) imply the existence of an involutive subdistribution of corank 1 in each X>W and, moreover, imply all regularity conditions X>W — T>i. It is important to observe that if conditions (i) and (ii) of the theorem are satisfied then the involutive subdistribution jCn-i is necessarily

80

unique (see Lemma 4.2), which allows to check easily condition (iii). In the case of Goursat structures, that is for m = 1, we give Theorem 5.7, an analogue of the above result, in which all information is, this time, encoded in the characteristic distribution Cn-\ of £>(" - 1 '. The paper is organized as follows. In Section 2 we define the canonical system for curves and we give Theorem 2.1, a first characterization of distributions that are locally equivalent to the canonical contact system for curves. In Section 3 we sketch briefly the history of results describing various canonical contact systems. In Section 4 we discuss the problem of whether a given distribution possesses a corank one involutive subdistribution and we recall Bryant's solution of that problem. Section 5 contains other characterizations of the canonical contact systems for curves. One of them, Theorem 5.4, is based on the notion of characteristic distributions while another, Theorem 5.6, uses only one involutive sudistribution in which all information about the geometry and about the regularity is encoded. The presented proof gives a method of constructing canonical coordinates for any distribution equivalent to the canonical contact system. Still in Section 5, we present an analogue of Theorem 5.6 for Goursat structures that uses, this time, only one characteristic distribution. Flatness of control systems equivalent to a canonical contact system for curves is discussed in Section 6. In particular, we show how the geometry of that class of systems helps to describe their flat outputs. We give a model of a mechanical nonholonomic system equivalent to a canonical contact system in Section 7. In Section 8 we introduce extended Kumpera-Ruiz normal forms and give a result, Theorem 8.1, stating that any contact system, regular or singular, is locally equivalent to an extended Kumpera-Ruiz normal form.

2

The Canonical Contact System for Curves

A rank k distribution V on a smooth manifold M is a map that assigns smoothly to each point pin M a, linear subspace V(p) C Tp M of dimension k. In other words, a rank A; distribution is a smooth rank k subbundle of the tangent bundle TM. Such a field of tangent fc-planes is spanned locally by k pointwise linearly independent smooth vector fields / i , . . . , fk on M, which will be denoted by X> = ( / i , . . . , / * ) . Two distributions V and V denned on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism ip between M and M such that ()(p) = V(p), for each point p in M. The derived flag of a distribution V is the sequence of modules of vector

81

fields D<°) C 2? (1) C • • • denned inductively by D(0)=£>

P< i + 1 ) = X>(i) + [ D ( i ) , P w ] ,

and

for t > 0.

(2.1)

The Lie /Jag is the sequence of modules of vector fields Do C T>\ C • • • defined inductively by V0=V

and

P i + i =Vt + [T>o,T>i], for t > 0.

(2.2)

In general, the derived and Lie flags are different; though for any point p in the underlying manifold the inclusion T>i(p) C X>W(p) clearly holds, for i > 0. For a given distribution V, defined on a manifold M, we will say that a point p of M is a regular point of V if all the elements T>i of its Lie flag have constant rank in a small enough neighborhood of p. A distribution V is said to be completely nonholonomic if, for each point p i n M , there exists an integer N(p) such that VN^(p) = TPM. A distribution V is said to be involutive if its first derived system satisfies V^ = £>(°'. An alternative description of the above defined objects can also be given using the dual language of differential forms. A Pfaffian system 1 of rank s on a smooth manifold M is a map that assigns smoothly to each point p in M a linear subspace l(p) c T*M of dimension s. In other words, a Pfaffian system of rank s is a smooth subbundle of rank s of the cotangent bundle T*M. Such a field of cotangent s-planes is spanned locally by s pointwise linearly independent smooth differential 1-forms w i , . . . ,ws on M, which will be denoted by I = (UJI,... ,w s ). Sometimes we will denote the Pfaffian system I = ( w i , . . . ,CJ S ) by ui = 0 , . . . , ws = 0. In the case of a rank one Paffian system I = (w) we will speak also about a Pfaffian equation u = 0. Two Pfaffian systems I and 1 defined on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism ip between M and M such that X(p) = (ip*l)(p), for each point pin M. For a Pfaffian system I , we can define its derived flag 1^ D 2' 1 ) D • • • by the relations I<°) = I and l( i + 1 > = {a e I ( i ) : da = 0 mod ! « } , for t > 0, provided that each element X^ of this sequence has constant rank. In this case, it is immediate to see that the derived flag of the distribution V = 1± coincides with the sequence of distributions that anihilate the elements of the derived flag of I , that is v(i)

=

(i«)-L ;

for i > o.

For a given Pfaffian system X, we will say that a point p of M is a regular point if p is a regular point for the distribution V =2±, that is if all elements T>i of the Lie flag are of constant rank in a small enough neighborhood of p.

82

Consider the space J n ( E , R m ) of jets of order n > 1 of functions from R into R m . This space is diffeomorphic to R ( " + 1 ) m + 1 . The canonical coordinates associated to R (denoted by x°) and to R m (denoted by x^,... , x ^ ) can be used to define the canonical coordinates on J n (R, R m ) , which will be denoted by 0 T° r° r1 r1 T" J-OJ-^-I)--- j i f n j J - j , . . . , x m , . . . , x j , . . . x

T"

,x

m

,

with obvious indentifications org = 9 and x° = p° = u*, for 1 < i < m, and xj = p3i, for 1 < i < m and 1 < j < n (see the beginning of the Introduction). Observe that any smooth map tp from R into R m defines a curve in J"(R, R m ) by the relations x{ = tpy\x°), for 0 < i < m and 0 < j < n, where ip^ denotes the j - t h derivative with respect to zfj °f the z'-th component of (p. This curve is called the n-graph of (p. It is clear that not all curves in J"(R, R m ) are n-graphs of maps. In order to distinguish the "good" curves from the "bad" ones, we should introduce a set of constraints on the velocities of curves in J n (R, R m ) . In other words, we should endow J"(R, R m ) with a nonholonomic structure. The canonical contact system on J n ( R , R m ) is the completely nonholonomic distribution spanned by the following family of vector fields: /

n—1 m

( d^f > • • • > 5 i ^ " » 9 4 + E T,xi

\

^7) •

(2.3)

In a dual way, we will also call the canonical contact system on J " ( R , R m ) the Pfaffian system (dx{ - x{+1dx%, 0 < j < n - 1, 1 < i < m) annihilating the above distribution. For the canonical contact system on J"(R, R m ) we will also use the name canonical contact system for curves. By definition, if a curve in J"(R, R m ) is the n-graph of some map then it is an integral curve of the canonical contact system. More precisely, a section a : R -> J n (R, R m ) is the n-graph of a curve ip : R -> RTO if and only if it is an integral curve of the canonical contact system on J"(R, R m ) (see e.g. 3 and 2 5 ) . The aim of our paper is to analyze various possible answers to the question "Which distributions are locally equivalent to the canonical contact system on J"(R, R m ) ? " . The following result, obtained by the authors 2 7 , will be the starting point of our study. Theorem 2.1 (regular contact systems, first version). A rank m + 1 distribution V on a manifold M of dimension (n + l)m + 1 is equivalent,

83

in a small enough neighborhood of any point p in M, to the canonical contact system on Jn(IR, E m ) if and only if the two following conditions hold, for 0^1' that has constant corank one in (ii) Each element T>i of the Lie flag has constant rank (i + \)m + 1. Remarks 1. This result yields a constructive test for the local equivalence to the canonical contact system for curves, provided that we know how to check whether or not a given distribution admits a corank one involutive subdistribution. We give in Section 4 a checkable necessary and sufficient condition for the existence of such a distribution. 2. Item (i) describes the essential geometric property of distributions equivalent to canonical contact systems for curves while the condition (ii) distiguishes regular points p at which X>'*'(p) = T^i(p) from singular points, where this last condition is violated. Different roles of (i) and (ii) justify Definition 2.2 below. 3. Theorem 2.1 was proved in 27 , as a special case of a more general result that gives a normal form for any system satisfying (i) (see Remark 2 above). In Section 5 we will present a direct proof of Theorem 5.6, which is another version, under the weakest possible assumptions, of Theorem 2.1. Definition 2.2 (contact systems). A rank m + 1 distribution T> on a manifold M of dimension (n + \)m + 1 is said to be a contact system if, for any 0 < i < n, each element T>^ of the derived flag, has constant rank (i + l)m + 1 and contains an involutive subdistribution Ci C X>W that has constant corank one in A contact system is said to be a regular contact system at p G M if it is, locally at p, equivalent to the canonical contact system on Jn(R, Rm). Otherwise it will be called a singular contact system at p e M. Theorem 2.1 implies that a contact system is a regular contact system at a point p if and only if the point p is regular. Other, than that of Remark 2 above, equivalent characterizations of regular points of contact systems will be given in Theorems 5.4 and 5.6 , and in Corollary 5.5. 3

History

In this section we give a brief histry of the following problem: Which Pfaffian systems are locally equivalent to the canonical contact system on the space Jn(Rk,Wn) of n-jets of maps from Rk to Mm ? This problem is a natural

84

generalization of the follwing classical problem stated in 1814 by Pfaff: Which Pfaffian systems are locally equivalent to the canonical contact system on the space J1 (Rk,Rm) of 1-jets of maps from Rk to R? Studies of that problem have mainly concentrated on the four following cases of equivalence to the canonical contact system on the space • J^Rfc.R): Pfaff (1814), Frobenius (1877), and Darboux (1882); • J"(M,M): Engel (1890), E. von Weber (1898), Cartan (1914), Goursat (1923), Giaro, Kumpera and Ruiz (1978), Libermann (1978), Murray (1994), Mormul (1998-2001), Pasillas-Lepine and Respondek (1998-2001), Montgomery and Zhitomirskii (1999), • J ^ I f ^ i r ) : Gardner (1972), Bryant (1979); • Jn(R,Rm): Garnder and Shadwick (1992), Murray (1994), Tilbury and Sastry (1995), Aranda-Bricaire and Pomet (1995), Pasillas-Lepine and Respondek (1999-2001). In the following subsections we will describe solutions in each of the four above cases. Throughout this section we will use the notation p° = Uj, for 1 < i < m (see Introduction). 3.1

Pfaff-Darboux normal form

In this subsection we recall the classical solution given by Frobenius and Darboux to the original Pfaff question: which Pfaffian systems are locally equivalent to the canonical contact system on the space J 1 (Rk, E) ? We will need the following notion introduced by Frobenius n and Darboux 7 Definition 3.1. The class, at a point p, of the Pfaffian equation u = 0 is the largest integer p such that

Using the notion of class, Darboux 7 gave the following celebrated answer to the problem. Theorem 3.2 (Darboux, 1882). A Pfaffian equation w = 0, defined on a manifold of dimension 2k + 1, is locally equivalent to the equation given by the following canonical contact form dp0 - pldqi

phdqk = 0

on J1(Rk, R), if and only if its class is constant and equals k. The above canonical contact form is called Pfaff-Darboux normal form.

85

3.2

Bryant normal form

In this subsection we will recall a solution to the generalization of Pfaff's problem to the space of 1-jets of arbitrary maps. The following question has been studied by Gardner since 1972, and by Bryant (who gave a solution in 1979): which Pfaffian systems are locally equivalent to the canonical contact system on the space Jl(Wk, W1) ? In order to give an answer we will need the following generalization of the notion of class of a Pfaffian equation. Definition 3.3. The Engel rank, at a point p, of a Pfaffian system I — [uii,... ,uis) is the largest integer p such that there exists a 1-form a inX for which {{da)p A w i A - - - A w s ) ( p ) / 0. Obviously, in the case of a single Pfaffian equation u = 0, the notion of Engel rank reduces to that of class. A characteristic vector field of a distribution P is a vector field / that belongs to V and satisfies [/, V] C T>. The characteristic distribution of V, which will be denoted by C, is the module spanned by all its characteristic vector fields. It follows directly from the Jacobi identity that the characteristic distribution is always involutive but, in general, it needs not be of constant rank. If for a constant rank Pfaffian system I , the characteristic distribution C of the distribution V = l1- is of constant rank, then its anihilator (C)1is a Pfaffian system, called the Cartan system of the Pfaffian system I and denoted by C(X). We refer the reader to 3 for a definition of the Cartan system given in the language of Pfaffian systems. The following result was proved by Bryant in his Ph.D. Thesis 2 (see also 3 ) . Theorem 3.4 (Bryant, 1979). Assume that m ^ 2. A Pfaffian system 2=

(oJi,...,ojm),

defined on a manifold of dimension m + k + mk and such that I^1' = 0, is locally equivalent to the canonical contact system p\dqk = 0

Pm ~ Pmdqi

Pkmdqk = 0

d

dp\ - p\dqi

on J1 (Rk, lRm), if and only if its Engel rank is constant and equals k, and the rank of its Cartan system is constant and equals m + k + mk. The above normal form of a Pfaffian system is called Bryant normal form. Reasons for which the case m = 2 is excluded will be explained in Section 4, see Lemma 4.2 and Corollary 4.3.

86

3.3

Goursat normal form

Starting with the work of Engel 8 and von Weber 33 , the following question has been intensively studied: which Pfaffian systems are locally equivalent to the canonical contact system on the space J"(R, R) ? The case n — 1 was solved by Pfaff in his pioneering paper 29 ; it is also the simplest case of Darboux theorem given above. The case n = 2 was solved by Engel 8 . In order to give a solution for an arbitrary n, recall, see Introduction, the following fundamental notion. The derived system I' of a Pfaffian system 1 = (UJI, ...,u!s) is the Pfaffian system generated by all 1-forms a in I such that da AOJI A • • • A ws = 0.

Starting from a Pfaffian system 1 and its (first) derived system I ' we define the flag of derived systems 1^ D 1^ D • • • D 1^ D • • • inductively by I<°) = J a n d I < i + 1 ) = ( ! « ) ' . The following characterization of Pfaffian systems, that are locally equivalent to the canonical contact system on the space J"(R, R), was found by von Weber and has been intensively studied by Cartan. Theorem 3.5 (van Weber, 1898 ; Cartan, 1914). A Pfaffian system X = (uii,...,LJn) of rank n, defined on a manifold of dimension n + 2, is is locally equivalent (on an open and dense subset) to the canonical contact system dp0 - pldq = 0 , • • • , dpn~l - pndq = 0 on Jn(M., E) if and only if the elements 1^ of the flag of derived systems are of constant rank n — i, for 0 < i < n. The canonical form present in the above theorem is called Goursat normal form, mainly because it was intensively used by Goursat in his book 14 . We will call a Goursat structure any distribution V (resp. any Pfaffian system I) on a manifold of dimension n + 2 such that the elements of the derived flag X>(") are of constant rank i + 2 (resp. the elements j M 0 f the flag of derived systems are of constant rank n — i, for 0 < i < n). Therefore the above theorem says that a Goursat structure is locally equivalent to a Goursat normal form on an open and dense subset of the underlying manifold. Notice, see Proposition 5.1, that a Goursat structure is just a contact system of rank 2 (a contact system of corank 2, if we use the language of Pfaffian systems). Theorem 3.5 is thus a special case of a more general Theorem 2.1 stating that a contact system is equivalent to the canonical contact system on an open and dense subset. Applying the latter theorem to the case of Goursat

87

structures, the open and dense subset of Theorem 3.5 can be described as points p at which X>W(p) = T>i(p) (as shown earlier by Murray 24 directly for Goursat structures). Other equivalent descriptions of the open and dense set of regular points of Goursat structures will follow from characterizations of regular points of contact systems given in Theorems 5.4 and 5.6 and in Corollary 5.5, stated in Section 5. Here we would like only to emphasize that in the cases n = 1 and n = 2 we do not have to restrict ourselves to an open and dense set. In fact, the conditions dim X^\q) — 0, in the case n = 1, and dim l{1)(q) = 1, dim l(2)(q) = 0, in the case n = 2, satisfied for any q in a neighborhood of a given point p, guarantee that the Pfaffian system I on a manifold, respectively of dimension 3 and 4, is locally equivalent around p to Goursat normal form. Giaro, Kumpera, and Ruiz 13 were the first ones who observed that starting from n = 3, singularities may appear for Goursat structures. Example 3.6 (Giaro et al, 1978). The following Pfaffian system of five variables dx2 — xzdxi = 0 dx$ — Xidxi = 0 dx\ — x^dxi = 0 is a Goursat structure, that is, it satisfies the conditions of the above theorem but is not locally equivalent to the canonical contact system on J 3 (E, E) if x5=0. It was this example with which an increasing interest in the problem of singularities of Goursat structures started (for more results see e.g. 17 , 6 , 22 , 23

28 \

21

!

3.4

>

I'

Canonical contact systems for curves

In this section we will discuss the following question: which Pfaffian systems are locally equivalent to the canonical contact system on the space J"(M, E m ) , that is, to the canonical contact system for curves? We will put the answer, given to that question as Theorem 2.1 in Section 2, in a form analogous to that of Theorems 3.4 and refthm-goursat. Theorem 3.7 (regular contact systems, second version). A Pfaffian system I of constant rank nm, where m ^ 2, defined on a manifold of dimension (n + \)m + 1, is locally equivalent (on an open and dense subset)

88

to the canonical contact system dp0! - p\dq = 0 , • • • , dp7?'1 - p^dq = 0 dp°m -pldq

= 0 , • • • , dp^-1 -p^dq

=0

on J"(E, R m ) if and only if each element j W of the flag of derived systems satisfies the following conditions, for 0 < i < n. (i) 1^ is of constant rank (n — i)m; (ii) The Engel rank ofl^ equals one; (iii) The rank of the Cartan system of I^1' is constant and equals (n — i + l)m + l. The above theorem is given, as Corollary 2.4, in 2 7 . Its proof follows directly from Theorem 2.1 and a characterization, via the notion of Engel rank, of the existence of an involutive subdistribution of corank one in a given distribution. Such a characterization was obtained by Bryant 2 and will be discussed in Section 4. Notice that the formulation of Theorem 3.7 can be considered as a natural combination of, on the one hand, Weber-Cartan's Thereom 3.5 and, on the other hand, of Bryant's Theorem 3.4. Indeed, like Thereom 3.5, it involves the derived flag and requires to exclude some singular points. On other hand, like Theorem 3.4, it expresses the basic geometry of the problem through the notion of Engel rank. As we said, we will discuss links between the geometry and Engel rank in Section 4. What concerns regular points, one characterization as points p such that V^ (p) = T>i (p) was already given in Theorem 2.1. We would like, however, to express that regularity condition in terms of the Pfaffian system I ; but is not clear how to represent the elements T>i of the derived flag in terms of X. For that reason, we will present in Section 5 other equivalent descriptions of regular points, in particular, in terms of Pfaffian systems. 4

Involutive s u b d i s t r i b u t i o n s of corank one

The aim of this Section is to provide checkable conditions to verify (i) of Theorem 2.1 and to justify the statement of Theorem 3.7. This will be done by answering the following question: "When does a given constant rank distribution V contain an involutive subdistribution C C T> that has constant corank one in T>T\ In fact, the answer to this question is an immediate consequence of a result contained in Bryant's Ph.D. thesis 2 . Links between Bryant's result and the characterization of the canonical contact system for

89 curves have also been observed by Aranda-Bricaire and Pomet 1. Recall that in Subsection 3.2 we defined the Engel rank of a Pfaffian system X, at a point p, as the largest integer p such that there exists a 1-form u in X for which we have (dw)p(p) ^ OmodX. The Engel rank of a constant rank distribution V will be, by definition, the Engel rank of its anihilator V-1. Obviously, the Engel rank p of a distribution equals zero at each point if and only if the distribution is involutive. We will now give an equivalent definition of the Engel rank, see 2 7 , in the language of vector fields, in the particular case when p — 1, which will be important in the paper. Let V be a distribution such that Z>(°) and T>^ have constant ranks d0 and di, respectively, and denote ro(p) — d\ (p) — do(p). Assume that d0 > 2 and r 0 > 1. Take a family of vector fields ( / l , - - • , / d 0 ) 5 l ) - - - >#r 0 )

such that 2>(°) = ( / ! , . . . , / d o ) and V& = ( A , . . . ,fdo,gu . . . ,gro). The structure functions ck- associated to those generators, for 1 < i < j < d0 and 1 < k < ro, are the smooth functions defined by the following relations: [/* . / > ] = £ 4 9k mod P<°>,

for 1 < i < j < d0.

It is important to point out that the structure functions are not invariantly related to the distribution V, since they depend on the choice of generators. Assume that the Engel rank p of D is constant and that ro > 1. It is easy to check that p = 1 if and only if either do = 2, or do = 3, or do > 4 and the structure functions satisfy the relations

4-c« - <44 + &U + %A - ,& + « , • = o-

(4-i)

for each sextuple (i,j,k,l,p,q) of integers such that l 1. Then the two following conditions are equivalent: (i) The characteristic distribution CofT> has constant rank CQ = do—ro — 1 and the Engel rank p of V is constant and equals 1; (ii) The distribution T> contains a subdistribution B CD that has constant corank one in T> and satisfies [B,B] C T>.

90

Observe that if the first condition is satisfied then we must necessarily have r 0 < do — 1. The following result is included in the proof of Bryant's 2 normal form Theorem (see also 3 ) . An alternative proof of its surprising item (iii) is given in Appendix A of 27 , where the role of the assumption ro > 3 is explained by relating it to the Jacobi identity. Lemma 4.2 (Bryant). LetV be a distribution such thatV^ andV^ have constant ranks do and d\, respectively. Assume that the distribution T> contains a subdistribution B C T> that has constant corank one in T> and satisfies

[B,B]cV. (i) If ro = 1 then the distribution T> contains an involutive subdistribution C C T> that has constant corank one in T>; (ii) Ifr0>2 then B is unique; (iii) If ro > 3 then B is involutive. Observe that, in the first item of the above Lemma, the involutive subdistribution C can be different from B, which is not necessarily involutive. The following result is a direct consequence of Bryant's work. In order to avoid the trivial case ro = 0, for which the existence of a corank one involutive subdistribution is obvious, we will assume that r 0 > 1. Corollary 4.3 (corank one involutive subdistributions). Let V be a distribution such that X>'0' and T>^ have constant ranks do and d\, respectively. Assume that ro > 1. Then, the distribution V contains an involutive subdistribution C C T> that has constant corank one in D if and only if the three following conditions hold: (i) The characteristic distribution C ofT> has constant rank Co — do—ro — 1; (ii) The Engel rank pofDis constant and equals 1; (iii) If ro = 2 then, additionally, the unique corank one subdistribution B C T> such that [B, B] C D must be involutive. We would like to to emphasize that the above conditions are easy to be checked, and, as a consequence, so are the conditions of Theorems 2.1 and 3.7. Indeed, for any distribution, or the corresponding Pfafnan system, we can compute the characteristic distribution C and check whether or not the Engel rank equals 1 using, respectively, the formula (4.2) and the condition (4.3) below. This gives the solution if ro ^ 2 (equivalently, if m / 2). If r 0 = 2 we have additionally to check the involutivness of the unique distribution B satisfying [B, B] C V, whose explicit construction is also given by (4.4) below. In the sequel, we will show, following Bryant 2 , how to verify whether or not the Engel rank equals 1, and how to construct explicitly — if it exists — the unique corank one subdistribution B C V satisfying [B, B] C V (which, in the case r 0 > 2, is the unique involutive subdistribution of corank one in V).

91 Consider a distribution T> of constant rank d0, defined on a manifold of dimension N. Let w i , . . . ,w So , where so = N — do, be differential 1-forms locally spanning VL, the annihilator of V, that is V± =

(ui,...,uao).

We will denote by I the Pffafian system generated by u\,..., For any form ui 6 V-1, we put W(w) = {/GZ> :

wSo.

/jdweP1}.

Clearly, the characteristic distribution C of V is given by C=f)W(oJi).

(4.2)

Now assume that V^ is of constant rank di > do, that is ro > 1, or, equivalently, that the first derived system 2^ is of constant rank smaller than soBy a direct calculation we can check (see e.g. 3 ) that the Engel rank of the distribution V, or of the corresponding Pfaffian system I, equals 1 at p if and only if (dui A db}j)(p) = 0

modi,

(4.3)

for any 1 < i < j < soNow let us choose a family of differential 1-forms LJi,...,u)ro,Wr0+i,...,uSo such that I = (V^)1= (OJX,. .. ,wSo) and i W = (D^)1= (uv 0 +i,... ,w So ). When condition (i) of Lemma 4.1 is satisfied, independently of the value of ro > 2, the unique distribution B satisfying [B, B] C V is given, as shown by Bryant 2 , by B=Y:\V(LJi).

(4.4)

i=l

In fact, Bryant has also proved that it is enough to take in the above sum only two terms corresponding to any 1 < i < j < ro. In order to verify, in the case ro = 2, the conditions of Corollary 4.3 we have additionally to check the involutivity of this explicitly calculable distribution. Notice that, if the distribution V satisfies Lemma 4.1 (i) and r 0 > 3, then the formula (4.4) gives the unique involutive subdistribution C of corank one in V.

92 5

5.1

Contact systems, characteristic distributions, and involutive subdistributions Contact systems and characteristic

distributions

Let V be a contact system, that is, according to Definition 2.2, each distribution Z?W contains a unique involutive subdistribution d of corank one. It turns out that those involutive subdistributions are unique and, moreover, can be expressed in terms of characteristic distributions. Indeed, we have the following result. Proposition 5.1. Let T> be a rank m + 1 contact system, defined on a manifold of dimension (n + l)m + 1. Each distribution P ^ ' — either for 0 < i < n — 2, if m = 1, or for 0 2 — contains a unique involutive subdistribution Ci C P ' 4 ' that has constant corank one in P « and that, for 0 < i < n — 2, coincides with the characteristic distribution Cj+i o/p( J + 1 >. The following picture summarizes the above proposition. £>(°) c P ^ C • • • C X>("-2) C Pf"- 1 ) C P ( n ) = TM U U U U Co C C\ C • • • C Cn-2 C Cn-\ n n ii C\ C C2 C • • • C C n - i

n £>(i)

n c

(5.1)

n

£>(2) C • • • C D^""1)

The inclusions between corresponding successive elements of the first and second row are those of the definition of contact systems. The inclusions between the elements of the third and fourth row express simply the fact that Ci, being a a characteristic distribution of V^\ is contained in 2?W. The equalities between the corresponding elements of the second and third row express a basic property of the sequence of involutive subdistributions Ci. In the case m = 1, that is for Goursat structures, Proposition 5.1 seems to be already known to Cartan 5 . For a proof we refer the reader to Montgomery and Zhitomirskii 21 (see also 4 , 17 , and 2 0 ) . Incidence relations of distributions V^'s and Cj's have been intensively used by the authors 28 and by Montgomery and Zhitomirskii 21 to study the geometry of Goursat structures at singular points. In the case m > 2, to the best of our knowledge, Proposition 5.1, has never been stated. Notice a substantial difference between the cases m = 1 and m > 2, which is the presence of the canonical involutive subdistribution Cn-i in the second row for m > 2. In the case of Goursat structures, that

93 is if m = 1, the distribution p ( " - 1 ) contains many different involutive sudistributions of corank one but non of them has any canonical properties and therefore in the case m = 1, the second row of (5.1) terminates with £„_2This basic difference between the cases m = 1 and m > 2 will be reflected in different statements of Theorems 5.6 and 5.7, that describe, respectively, regular contact systems for m > 2 in terms of £„_i and, for m = 1, in terms of C n _i = £„-2Definition 5.2. We will call the corank one involutive subdistribution d given in the previous result — either for 0 < i < n — 2, if m = 1, or for 0 2 — the canonical subdistribution o / p W . When proving Proposition 5.1 we will use the following result proved in 2 7 . Lemma 5.3 {Co C CI). Let T> be a distribution such that p ( ° ' , T>^\ and T>W have constant ranks do, di > do + 2, and c?2 > d\ + 2, respectively. Assume that each distribution p W , for i = 0 and 1, contains an involutive subdistribution Ci C P ^ that has constant corank one in p W . Then Co CCiProof: (of Proposition 5.1). In the case m = 1, that is for Goursat structures, the result was proven in 21 (see also 4 , 17 , and 2 0 ) . Consider the case m > 2. By Lemma 4.2, we conclude that all involutive distributions d, for 0 < i < n—1, are unique. In order to show that d = Cj+i, for 0 < i < n — 2, apply Lemma 5.3, with T>(°\ T>^1\ and V^2\ being replaced, respectively, by V^\ T>(i+1\ and X>('+2). We conclude that d C Ci+1. On the one hand, the rank of d equals m{i +1) while, on the other hand, the rank of Ci+i is also equal to m(i +1), because V^+1^ contains a subdistribution C,+i that satisfies [Ci+i,d+i] C X>('+1) (see Lemma 4.1). We thus have d = Ci+iD In the case of Goursat structures, that is m = 1, using the canonical subdistributions we can distinguish singular points from regular ones, see 21 and 2 8 . Below we show that the geometry of incidence of canonical subdistributions describes completely also regular contact systems in the general case m > 2. Indeed, we have the following result. Theorem 5.4 (regular contact systems, third version). A rank m + 1 distribution T>, with m > 2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of a point p in M, to the canonical contact system on J"(R, E m ) if and only if the following conditions hold. (i) Each element P ^ of the derived flag, for 0 < i < n, has constant rank (i + \)m + 1. (ii) Each element T>^> of the derived flag, for 0 < i < n — 2, contains the characteristic distribution C,+i of V^l+1' that has constant corank one in P « .

94 (iii) X>(™-1) contains an involutive subdistribution £n-i that has constant corank one in Z)( ra_1 ). (iv) For any 0 < i < n — 3, we have 1)W (p) is not contained in Ci+2 (p) and, moreover, X>(n_2)(p) is not contained in £„_i(p). Remarks 1. Let us look at items (i)-(iv) in the case of of Goursat structures, that is m = 1. Firstly, item (i) implies, according to Proposition 5.1, item (ii). Secondly, in the case m = 1, as shown in 21 and 2 8 , a Goursat structure (that is a distribution of rank two satisfying item (i)) is locally at p equivalent to Goursat normal form under the regularity conditions V^l\p) ^ d+2(p), for 0 < i < n — 3. Thirdly, being equivalent to Goursat normal form, it clearly satisfies (iii) because it admits infinitely many involutive subdistributions of corank one in Z>(™_1), most of which fulfilling the second part of item (iv). 2. Notice that any distribution T> satisfying (i), (ii), and (iii) is a contact system. Indeed, since the characteristic distributions C; are involutive, we put d = Ci+i and thus each distribution 2?W, for 0 < i < n — 2, contains an involutive distribution £j C V^ of corank one because of (i) and (ii). The one before the last distribution 2?(™_1) contains, by (iii), an involutive subdistribution £ n - i , which is unique and it is the canonical involutive subdistribution in P ^ - 1 ) . 3. As we argued in Remark 2 above, (i)-(iii) imply that T> is a contact system. Item (iv) is thus another way of expressing the regularity condition 2?W(p) = T>i(p). Contrary to this last condition, however, item (iv), as well as items (i)-(iii), can be immediately expressed in terms of Pfaffian systems, which we will state below as Corollary 5.5. 4. We will omit a proof of Theorem 5.4 because it follows from a more general result, Theorem 5.6, which will be stated in Subsection 5.2 and proven in Subsection 5.3. Corollary 5.5 (regular contact systems, fourth version). Let I be a Pfaffian system of rank nm, defined on a manifold M of dimension (n + l)m + 1. If m ^ 2, the Pfaffian system I is locally equivalent, at a given point p of M, to the canonical contact system on J n (M, E m ) if and only if (i) The rank of each derived system ZW is constant and equals (n — i)m, for 0 < i < n. (ii) The rank of each Cartan system C(I^) is constant and equals (n + 1 — i)m + 1, for 0 < i < n — 1. (iii) The Engel rank ofl^n~^ is constant and equals 1. (iv) For any 0 < i < n - 3, C(l^+2^){p) is not contained in l^{p) and, moreover, Jn-\{p) is not contained in T^n~2^[p), where Jn-\ is the unique integrable Pfaffian system such that Z(™_1) is of constant corank

95

one in Jn-\, and whose existence is given by (ii) and (iii). Notice two differences between Corollary 5.5 and Theorem 3.7. Firstly, as we have mentioned in Remark 3 just above, item (iv) of Corollary 5.5 gives a description of the open and dense set of regular points of Theorem 3.7. Secondly, when using Corollary 5.5 we have to calculate the Engel rank of l ' " - 1 ) only (equivalently, to verify the existence of an involutive subdistribution of corank one in the distribution X>(™_1) only). 5.2

Contact systems and involutive subdistribution

£n-i

The aim of this subsection is to show that all information about a distribution V, which is locally equivalent to the canonical contact system on J n (K, R m ) , that is, about a regular contact system, is encoded completely in the canonical involutive subdistribution £ n _ i . This information concerns, simultaneously, the existence of all canonical distributions d = C;+i, for 0 < i < n — 2, the regularity of the point p, as well as the desired ranks of all intermediate distributions 2?W. In fact, we have the following result. Theorem 5.6 (regular contact systems, fifth version). A distribution V of rank m + 1, with m >2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of a point p in M, to the canonical contact system on Jn(R, K m ) if and only if the following conditions hold. (i) £>(") = TM. (ii) XH" -1 ' is of constant rank nm + 1 and contains an involutive subdistribution £n-i that has constant corank one in (iii) T>(°)(p) is not contained in Cn-i{p). If the conditions (i)-(iii) are satisfied then, of course, the distribution £ „ _ i , whose existence is claimed by (ii), is unique and it is with the canonical subdistibution of p f " " 1 ) . It is interesting to see how much information is encoded just in the existence of the canonical involutive subdistribution Cn-i and its regular intersection with Z>(°). In this case, the distribution £n-i "knows" about the existence of all canonical distributions £, = Cj+i, as well as about the regularity of all intersections of Z>(*) with C i + 2 . Notice, however, that although Cn-\ "knows" how to distinguish regular points from singular ones, it does not "know" about the geometry of the incidence of various DW's and C,-'s at singular points corresponding to different extended Kumpera-Ruiz normal forms (see Section 8). Theorem 5.6 was stated for m > 2, thus excluding m = 1, for the obvious reason: in the case of Goursat structures, that is for m = 1, we do not have the

96 canonical involutive subdistribution £ „ - i . The second row of (5.1) terminates at £„_2 = Cn-\. It can be noticed, however, that a "non-canonical" version of Theorem 5.6 holds also for m = 1: a rank-two distribution is equivalent to a Goursat normal form if and only if there exists a distribution £„_i satisfying the conditions (i),(ii), and (iii) of Theorem 5.6. Necessity is seen immediately from Goursat normal form while sufficiency follows from the sufficiency part of the presented proof of Theorem 5.6, which is the same independently of the value of m. What is important to emphasize, however, is that in the case m = 1, the condition (iii) is, in general, non veryfiable. In fact, if ranks of D( n _ 1 ) and T>^ equal, respectively, n + 1 and n + 2 then X>("-1) contains an infinity of involutive subdistributions of corank one. In order to check condition (iii) we have to calculate (some of) them and it is not clear how to do it because the method presented in Section 4 and based on Engel rank does not apply. Moreover, even if we would be able to find one such an involutive subdistribution, it may happen that it does not satisfy (iii) while others do. We will show now that an analogue of Theorem 5.6 holds for Goursat structures, that is for m = 1, with Hn-\ replaced by Cn-\. In fact we have the following result. Theorem 5.7 (Goursat normal form). A distribution T> of constant rank two, on a manifold M of dimension n + 2, is equivalent, in a small enough neighborhood of a point p in M, to Goursat normal form, that is to the canonical contact system on J n (R, E), if and only if the following conditions hold. (i) £>(") = TM. (ii) J)(n~2> is of constant rank n; moreover, the characteristic distribution Cn-i of X>(ra_1) is contained in £>(" -2 ) and has constant corank one in T>(n~2K (iii) p(°)(p) is not contained in Cn-i(p). Notice that we assume neither that T>^, for 0 < i < n — 3, are of constant rank nor that X>(n_1) is of constant rank nor that the ranks of 2?W grow by one at any point: the characteristic distribution C n _i "knows" about all that. 5.3

Proof of Theorem 5.6

Proof: Necessity is obvious. We prove sufficiency by giving a sequence of local diffeomorphisms that bring the distribution V to the canonical contact system on J"(E, R m ) . For simplicity, we will use the notation

dx>'

K

dx{,'"'dxL

97

Since £ n - i is an involutive distribution of constant corank m + 1 in TM we can choose m + 1 smooth functions o, <j>i, . . . ,0 m such that (d(j)0,d(l)i,...,d(f>m) = (^n-i)- 1 in a neighborhood of p. Item (iii) implies that there exists a vector field g~o € T>(°> such that go(p) £ £n-i(p). Moreover, there exists a function fa such that LgQ0(p) ^ 0. Define 1 Introduce coordinates x% = fa, and x° = 0j, for 1 < i < m, and complete them to a coordinate system (x^,x°,... ,xn), where xj = (x[,... ,x3m), for 0 < j < n. Denote \ = Lgo4>°, for 1 < i < m. We have

r

- JL

( ~ - - dz1'"m'dx»)

J n 1 (

A-)

and d ''O

^

A l

d

,. d

d .

i=l

Since p(™ ^ is of constant rank and corank of £ n _ i in 2?(" ^ equals one, item (iii) implies that 2?( n_1 ) = £ „ _ i + (#0)- Hence Z>(") = pf"" 1 ) + [ZXn-^ZX"-!)] is given by X > ( n ) = £ „ _ i + [9d,£„_i] + (S0). This relation and item (i) imply that we can choose among - ^ j , for 1 < i < m, 1 < j < n, vector fields / i , . . . , fm, for such that P<"> = £ „ _ i + ([po, /i], 1 < » < m) + (go). In particular, this implies that the differentials dfa^,, d\, for 0 < j < 1 and 1 < i < m, are linearly indpendent at p. By relabelling x\, for 1 < i < m, 1 < J < n i if necassary, we can assume that /j = g|r, for 1 < J < m. Introduce new coordinates by replacing xj by fa\. In new coordinates, denoted still by x\, we have

98

and d

V^

i

d

.,

d

d ^

We claim that

Suppose that there exists a vector field / £ X>(n_2) such that / ^ J"„-2- Since go € D("-2> and P< n - 2 ) c X*^"1) = ( 5o ) + £ „ - i , we can suppose that

for suitable smooth functions /3j, 1 < i < m, where at least one function, say /3k, is not identically zero in a neighborhood of p. Since / € X>(n_2) and 2 but [/, 5o ] = E " i Agftr ffo G Z><°) C P ( " - ) , it follows that [f,g0] € V^V _1 mod£„_!, which clearly is not in X>(™ ) = £ n _ i + (g0) thus yielding a contradiction. Hence it follows that V(n~2) C ( § § * , . . . , ^ ) + (
A =4>\ iorO<j
=Lgoi~1,

— l and 1 < i < m. Suppose, moreover, that in those coordinates

as well as 2 ? ( " - * ) c ^ „ _ ? ) = ( ^ , . . . , ^ ) + («*,)

(5.3)

and ^-

+ 1 )

=(5o)

+

(^zr)mod(A,...,^_).

(5.4)

Notice that we do not assume the ranks of £>(""«) and of 1)("-
99

We will prove that equations (5.2), (5.3), and (5.4) hold with q being replaced by q + 1. To start with, observe that X>("-«+1) = Z>(n-«) + [£>("-»),£>("-«)] together with (5.3) imply that we can find m vector fields h, • • •, fm G X» (n_9) such that P<"-' + 1 > = &>) + ([/<,»>], l < i < m )

mod(^,...,^).

After relabelling xj, for q < j < n and 1 < i < m, if necessary, we can suppose, because of (5.3) that d d d t A< ^ /i = ^ ™ ° d ( ^ T I , . . . , — ) .

G T>(n~9^, the above expression implies that

Since fi,...,fm ^

^

M

+ ^ m o d ^ , . . . , ^ ) .

(5.5)

Moreover, it follows that the differentials d(j>Q, dcjyf, for 0 < j < q and 1 < i < m, are linearly indpendent at p. Introduce new coordinates by replacing x\ by

for 1 < i < m. In new coordinates, denoted still by x\, the distribution (— — ) v d a ; « ' ' " ' dxn' is preserved while the vector field g0 becomes

for some smooth functions 4>J+1, 1
<m.

T

, . . . , A ) + (»,).

(5.7)

Suppose that there exists a vector field / e X>("-«_1) such that / ^ ^ r ( n _,_ 1 ). Since y0 G £>("-(""«), we can suppose, because of (5.5), that ^

Q

Q

Q

100

for suitable smooth functions /?,, 1 < i < m, where at least one function, say Pk, is not identically zero in a neighborhood of p. Since / e D t " - ? - 1 ) and g0 6 E H i f t ^ P (o) c 2?(n-9-i), it follows that [f,go] E 2>("-»> but [/, 5o ] = m o d ( g f 7 , . . . , gf^-), which clearly is not in p("-») because of (5.5). This yields a contradiction thus proving that (5.7) holds. Now observe that the three just proved conditions (5.6), (5.7), and (5.5) correspond, respectively, to (5.2), (5.3), and (5.4), with q being replaced by q + 1. Hence by an induction argument we arrive after n steps at Vi0)

C

*» =

(

}

^

+

(5o)

'

where r±

w=

n—1

m

p.

+

+

r\

a4 gS^ 'S|

mod( )

^-

Since r>(0) is of constant rank m + 1 it follows that X>(°) = T0 = ( ^ ) + (g0), which ends the proof. • 5.4

Constructing canonical coordinates

The presented proof gives a method to calculate a diffeomprphism bringing a regular contact system V of rank m + 1 to the canonical contact system on Jn(Rk, Rm), starting from m + 1 independent functions (j>%, 0 ° , . . . , £, whose differentials annihilate £ n _ i , that is

(d0g,d0;,...,d^) = (£n_1)-L. Choose a vector field gQ G V^ such that go(p) $ Cn-\. Without loss of generality (see the beginning of the proof), we can suppose that Lgo(f>Q(p) / 0. Define x° = <$ and

J

= &

L

9Q^i

=

for 0 < j < n and 1 < i < m. It follows from the presented proof that in the coordinates (XQ,X°, ..., xn), the contact system T> takes the form

T>-(JL

—

A

rv

i+'i)

101

5.5

Proof of Theorem 5.7

Proof: Necessity is obvious. In order to prove sufficiency, choose a local coordinate system (xi,... ,xn+2) such that x(p) = 0 £ R™+2 and that

c

-(d

-?-)

ax\

oxn-i

We will identify the point p with 0 € R"+ 2 . Since Z>(°) (0) is not contained in C„_i(0), choose a vector field g0 6 V^ such that gQ(0) ^ C„_i(0). By multiplying go, if necessary, by a nonvanishing function, we can assume that 9o = 9o 5I7 + 9o+1 asf^T + s^f^ m o d c « - i • Because C„_i is of corank one in X>("-2) and g0(0) $ C n _i(0), it follows that D<"- 2 ) = C n _! + (50). Hence

*~" = <»•£•••• • 5 b k ' s r ' - - ^ s ^ » -

(5 8)

-

We claim that dim Z>(n_1)(g) = n + 1 for any q in some neighborhood of p = 0. In order to prove it, we will firstly show that dim V^n~1'(q) < n + 1 for any q in some neighborhood of p = 0. If not, there is a sequence of points qi, i = 1,2,... such that qi -> p and such that dim V^n~1\qi) = n + 2. Take a sequence of open neighorhoods Ui, such that qi € Ui and dim Vl^n~1\q) = n + 2 for any q £ \J Ui. Hence on any Ui, the characteristic distribution C„_i of X^"" 1 ) coincides with D*""1) = TM, which contradicts the fact that C„_i is of rank n — 1. Now w are going to prove that dim V^n~l\q) = n +1 in a neighborhod of p = 0. Suppose that dim V^n~1\p) = n. Denote Cj = g | - , for 1 < j < n — 1. There exist an integer 1 < k < n — 1 and a sequence of points qt, i = 1,2,... such that qi -» p and such that [ck,9o](qi) £ 1>(n~2\qi); otherwise, in view of (5.8), we would have, in a neighborhood of p, that £>(™_1) = £>("~2), which contradicts V^ = TM. Take a sequence of open neighborhoods Ui, such that qi 6 t/j and [ck,go](l) $ X^ n _ 2 '(g) for any q &\JUi. From the fact that dim V^n~1\q) < n + 1 it follows that there exist smooth functions a_,-, for 1 < j < n — 1, defined on |J Ui and such that [CJ > 9o] = aj [cfc, go] mod 2?("- 2) .

(5.9)

on (JUi. Now recall that the vector fields Cj, for 1 < j < n - 1, are characteristic vector fields of Z>("_1) and hence [a, [cj,go]] 6 Z>' n_1 ), for any 1 < i, j , < n — 1. Thus we have 2>(n) = 2?(»-D +

( [ 5 o , [c

^

1 < j < n - 1).

102

Notice that (5.9) implies that [30, [9,So]] = ctj[g0, [ck,go]]mod'D<-n~1^ on \jUi. Since dim V^n-^(p) = n, it follows that dim X>(") = TM. We thus have proved that £>("-2) c p f " - 1 ) c £>(n) = T M is a sequence of distributions of constant rank equal to, respectively, n, n + 1, and n + 2. It follows, see e.g. 3 4 , 20 , 2 1 , 28 , that we can choose local ccordinates (a;i,...,ar n + 2 ) such that £>("-2) = (g0) + C„_i, where (g0) = arrf-igf^ + x « 9 ^ 7 + 9 i f n modC„_i and ax\

oxn-i

We define an involutive distribution £„_i = (gf~, • • • , gf-) and the remaining part of the proof follows the same line as that of Theorem 5.6. • 6

Flatness of contact systems

In this Section we will discuss flatness of nonholonomic control systems, that are feedback equivalent to a canonical contact system for curves. Consider two control systems, with m + 1 controls, m

S:

x — 2_]fi(x)ui

= f(x)u

and

(6.1)

i=0 m

t : x = ^2fi(x)

Ui = f(i) u,

(6.2)

i=0

evolving respectively on two open sets X and X of RN, where u = (u0,...,umy, u - (u0,...,um), / = (fo, • • •, fmf, and / = (fo,...,fm). We say that the systems E and E are feedback equivalent if there exits a smooth diffeomorphism ip : X -> X and a feedback u = /3u, where the matrix P, whose entries /3j are smooth functions, is invertible at any x 6 X, such that

MfP) = f, where for any vector field g on X we denote by
103

Since the distribution V remains invariant under any invertible transformation u = (iu, all objects that we construct with the help of V can be considered as feedback invariant objects attached to E. The concept of flatness was introduced to control theory by Fliess, Levine, Martin, and Rouchon 9 , 10 in order to characterize control systems that are linearizable by endogenous feedback. This concept goes back to Hilbert 1 5 and Cartan 5 It is easy to observe that contact systems are flat. In this section we explain how the geometry of contact systems, described in previous Sections, provides additional information on various aspects of the flatness of contact systems. Consider a nonlinear control system E of the form (6.1). We say that E is x-flat at (x*,u*) E M.N x E m + 1 (see 10 , 16 , 30 ) if there exits a neighborhood V0 = X0 x Uo of (x*,u*) and smooth functions (pt on X0, for 0 < i < m, called x-flat outputs, having the following property: there exits an integer k and smooth functions •ji, 1 < i < N, and Si, 0 < i < m, such that we have Xi = 7. (,¥>, •••,V (fe) ), Ui = Si(tp,tp,...,ip^),

l
where

• • •, 2; in the case m = 1, canonical contact systems for curves are Goursat structures and their flatness is well understood, see e.g. . Assume that the system E is feedback equivalent to a canonical contact system for curves. Then there exists an invertible feedback u = f3u such that the unique involutive subdistribution Co of corank 1 contained in I?' 0 ) = (/o, • • •, fm) is spanned by the m last columns of the matrix f(3. Any matrix (3 defining a feedback having this property will be denoted by /?,„„. By Ufnv{x) we denote the codimension one subspace of Rm+1 such that f(x)u € Co{x) for any u € Ufnv(x). It is clear that a control u belongs to Ufnv(x) if and only if the zeroth-component (uP~^v(x))° of the feedback modified control u = uf3^v(x) is identically zero. Put u°nv(x) = (u[3^v(x))°. Although f3inv is not unique, the expression for u°inv is uniquely defined up to multiplication by a nonvanishing function and its interpretation is clear: if we apply to the system E any control u such that u°inv = 0, then the system is governed by lRm -valued contol and all its trajectories remain tangent to the involutive sudistribution £0. It is u®nv which is crucial in dynamic linearization of E as shows the following result given in 26 Proposition 6.1 (flatness). Consider the control system E given by (6.1) and assume that it is feedback equivalent, in a neighborhood of x* € X, to the

104

canonical contact system on J"(K,M m ). Then we have. (i) £ is x-flat at any (x*,u*) such that (u*)°nv(x*) ^ 0. (ii) The dynamic precompensator zx

= z2 ;

Zn

=

(6.3) V0

linked with the system via the dynamic feedback law « — Hmv I

vfh

J J

where vm = ( « i , . . . ,vmY and 7 is a suitably chosen nonvanishing function, renders the overall system, controlled by {vo,vi,...,vm)t, static feedback linearizable at any point (x*,z*) such that z\ ^ 0. (iii) As x-flat outputs we can choose any m + 1 smooth functions (pa, ipi,..., (°) can be explicitely calculated (see 2 and 2 7 ) . Therefore we can calculate, up to multiplication by a nonvanishing function of the state, the single control which needs to be preintegrated: it is the control u°nv, which is the zeroth-component of the new control u = /3~^vu. Secondly, it is also the involutive distribution £ 0 that determines the controls at which the system ceases to be flat. Thirdly, the canonical subdistribution £ n - i , that is the last involutive distributions £ „ _ i in the sequence £o C • • • C Cn-\ determines flat outputs. We would like to stress that functions whose differentials aninihilate £ n _ i are not the only flat outputs of a system equivalent to the canonical contact system. They have, however, a certain minimality property. Indeed, when determining — with their help — the state and the control, we need to take the minimal possible number of derivatives. Equivalently, among all choices of (m + 1) smooth functions which, treated as pseudo-outputs, define a dynamically input-ouput linearizable system without zero-dynamics, a linearizing precompensator of the minimal dimension n is needed in the case of pseudooutputs whose differentials anihilate Cn-\-

105

7

A n Example

In this Section we will consider a simple example given by a kinematical model of a car for which both the front and rear wheels are steerable, see 31 . We denote by (xo,yo) (resp. by (xi,j/i)) the coordinates of the midpoint between the two front (resp. rear) wheels and by a0 (resp. by c*i) the angle of the front (resp. rear) axle with respect to the horizontal. Finally, by tp we denote the angle of the bar connecting the two axles with respect to the horizontal. The two nonholonomic constraints reflecting the fact that the front and rear wheels roll without sleeping are given, respectively, by fi0 = 0 and Qi = 0 where differential 1-forms ilo = sin aodxo — cos aodyo J7i = smaidxi — cosaidyo. The holonomic constraints %i —

XQ

+ c o s ij}

2/i = 2/o +sinV> define a map tp from (S 1 ) 3 x E 2 , equipped with the coordinates (tp,a0,ai,xo,yo) into (5 1 ) 3 x R 4 , equipped with the coordinates (ip,ao,a{^xo,,yo,xi,yi). The pull-backs wo =
= 7r/2 + kn, k 6 Z} C X

and

S2 = {ct\ — ao = kit, k € Z} C X. The differential 1-forms UJQ and uj\ are linearly independent at p 6 X if and only if p ^ S\ n S2- Therefore outside Si fl S2, the nonholonomic constraints define a control system of the form e = /o(0«o + / i ( 0 « i + / 2 ( 0 « 2 , where f = (^,oio,oii,xo,yo) are coordinates on X = (S 1 ) 3 x R 2 , and the distribution V = ( / 0 , / i , / 2 ) , where / 0 = -£-, h = 3^7, and / 2 = cos(ai - ip) ( c o s ( a 0 ) ^ + s i n ( a 0 ) ^ J + sin(ai - a 0 ) ^ .

106 It is interesting to notice that at points of S\ D S2, integral curves of the distribution V = (wo,Wi)x do not represent all trajectories of the system subject to the nonholonomic constraints (uo,uii). Indeed, through any point p £ Si f)S2 there pass trajectories of the system whose velocities do not belong to V(p). We thus consider the model of a car outside Si fl S2. Clearly, the distribution V contains the involutive subdistribution *

I da0 ' Sen / "

of corank one in V. Since n = 1, the distribution C = (~^-, ~^-) is the canonical involutive subdistribution £ n _ i . By a direct calculation we check that V^(p) is of rank five if and only if p £ Si U 53, where S 3 = {a0 - ip = TT/2 + fcTr, k € Z} C X. Therefore at any configuration point outside Si n S3, the model is feedback equivalent to the canonical contact system for curves on J X (E, R 2 ). It is interesting to observe that at points of S2 that are outside Si U S3, the system, although not being equivalent to a cononical contact system, is flat. It can, moreover, be transformed via a local feedback to the following normal form ±1 = x 5 u 0 ±2 =

X1U1

X3 =

U0

X4 =

Ui

x5 = u2, which generates a nilpotent Lie algebra. As flat outputs we can choose X2, X3, and Xi. Notice that the form of the set of singular controls at which the system ceases to be flat changes: it is formed by the union of the planes u0 = 0 and «i = 0 in E 3 . Finally, we would like to mention the following difference between kinematical model of a car towing trailers and that of a multisteerable car. The former defines a Goursat structure (that is a contact system of rank two) at any point of its congfiguration space and, conversly, any Goursat structure can be locally equivalent to that kinematical model at a well choosen point of its configuration space (see 28 and 2 1 ) . This correspondence is no longer valid for multisteerable cars. They define a contact system of rank m +1 > 3 at almost all configurations but, in general, they fail to do so at some configurations (like at p € Si U S3 for the considered example).

107

8

Singular points and extended Kumpera-Ruiz normal forms

The aim of this Section is to study the class of distributions that satisfy condition (i) of Theorem 2.1 but fail to satisfy condition (ii) of that theorem. The fist condition describes the geometry of the canonical contact system while the second condition characterizes regular points. Recall that a distribution that satisfies the former but fails to satisfy the latter at a point p is called a singular contact system at p (see Definition 2.2). In this Section we will recall a result of 27 stating that any such distribution can be brought to a normal form for which we propose the name extended Kumpera-Ruiz normal form. Those forms generalize the canonical contact system on J 1 (R, Rm) in a way analogous to that how Kumpera-Ruiz normal forms (see e.g. 17 , 2 1 , 2 3 , and 28 ) generalize the canonical system on J1(M,1R), which is also called Goursat normal form. Consider the family of vector fields K1 = (K\, ... , K^, KJ) that span the canonical contact system on JX(IR, M.m), where K1

i

- -AK1 - -Aaxp • • • ' "-m — aT£

—

and the family of vector fields AC2 = (K 2 , . . . , K^, KQ) that span the canonical contact system on J 2 (M, K m ) , where 2

-

K

a -A K-2 — H£[' • • • > ^m — o ^

"'l

—

K2

_ ji

a

,

, 2

a_ ,

a

I

,

,

d_ ,

I

a

Loosely speaking, we can write 2 9 d *2 K *"1 — 5 x f ' • • • ' ^m — a ^ T K0 = XxKt + |- XmKm + KQ.

In order to make this precise we will adopt the following natural notation. Consider an arbitrary vector field / given on Jn~1(R, E m ) by n —1 m

/ = E E / / ( ^ - 1 ) ^ + /o°(^-1)4> j=0i=l

u

i

where J " " 1 denotes the coordinates XQ,X°,... , x°m, x\,... , xlm,... , a;" - 1 ,... n_1 m , x ^ of J ( E , E ) . We can lift the vector field / to a vector field on Jn(R, Rm), which we also denote by / , by taking n—1 m f=

v w ^ - ^ j=0i=l

+^ n - ^ a

+Q

.

a +-.. + 0 - 4 - .

108

That is, we lift / by translating it along the directions -^,...

, -J^r-

Notation(lifts of vector fields) . From now on, in any expression of the form n% = S f c o a « ( x ) K r _ 1 ' t h e vector fields K ^ - 1 , . . . , / C ^ - 1 should be considered as their above defined lifts. Let K n _ 1 = ( K ? - 1 , . . . , K ^ - 1 , K ™ _ 1 ) denote a family of vector fields defined on Jn~1(M.,Mm). A regular prolongation, with a parameter c", of K n _ 1 , denoted by Kn = Rc„ ( K " _ 1 ) , is a family of vector fields nn = « , . . . , K £ , K£) defined on J " ( E , E m ) by „ra _

d

n

_

d

(8.1) where cn = ( c " , . . . , cJJ,) is a vector of m real constants. A singular prolongation, with a parameter c™, of /c n _ 1 , denoted by K™ = S C n (/e n - 1 ), is a family of vector fields Kn = (K?, . . . , KJJ,, K#) defined on J n (R, E™) by a

„n _

„n

«0" = (*? + W *

_

+ • • • + « - l + C_ 1 )«»-_ 1 1 + K""1 + XnmK%-\

n

^ " ^

where c = ( c " , . . . , c|J l _ 1 ,0) is a vector of m real constants, the last one being zero. A family of vector fields K" on J " ( E , R m ) , for n > 1, will be called an extended Kumpera-Ruiz normal form if K" = CT„ O • • • o ^ ( K 1 ) , where for each 2 < i < n the map Oi equals either Rct or Sci, for some vector parameters c1. In other words, a Kumpera-Ruiz normal form is a family of vector fields obtained by successive prolongations from the family of vector fields that spans the canonical contact system on J 1 (M, R m ) . The above defined prolongations and prolongations-based definition of extended Kumpera-Ruiz normal forms generalize for contact systems analogous operations introduced by the authors 28 for Goursat structures. Let x : M -> R(»+i)™+i =* J " ( R , R m ) be a local coordinate system on a manifold M, in a neighborhood of a given point pin M. We will say that an extended Kumpera-Ruiz normal form on J"(R, Rm), defined in x-coordinates, is centered at p if we have x(p) = 0. For example, on J 2 (R, E 2 ), we have the two following extended Kumpera-Ruiz normal forms -r2 d.

( JL-

-2,

\d^'

8if ) Xl 8^

I r2 +

a^

9

.

I -r1

9

.. I T 1

-+• X2 \X1 gjs

9

.. I

-TX2Q^

d "\ -+- Q^)

J ,

109

defined by R^^K1) and 5(0,o)(«*), respectively. These two normal forms are obviously centered at zero. Recall, see Definition 2.2 that we define a contact system as a distribution V such that each element 2?W of its derived flag has constant rank (i + l)m + l and contains an involutive subdistribution A C V^ that has constant corank one in PW, for 0 < i < n, that is as a distribution satisfyng the condition (i) of Theorem 2.1. Moreover, we say that a contact system is regular at p if it satisfies at p the regularity condition (ii) of that theorem and is singular at p if it fails to satify that condition. The following theorem from 27 asserts that extended Kumpera-Ruiz normal forms serve as local normal forms for all, regular or singular, contact systems for curves. Theorem 8.1 (extended Kumpera-Ruiz normal forms). A contact system T> of rankm+1 on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of any point p in M, to a distribution spanned by an extended Kumpera-Ruiz normal form, centered at p and defined on a suitably chosen neighborhood of zero. Acknowledgments This survey is based on lectures delivered by the authors at the Conference on Geometric Control Theory and Applications, organized by Alfonso Anzaido Meneses, Bernard Bonnard, Jean-Paul Gauthier, and Felipe Monroy Perez, at Universidad Autonoma Metropolitana (Unidad Azcapotzalco), Mexico City. The authors are grateful to the local organizers, in particular to Alfonso Anzaido Meneses and Felipe Monroy Perez, for having organized the conference and for their hospitality. Moreover, they are grateful to Jean-Baptiste Pomet for discussions on his results 1. References 1. E.Aranda-Bricaire and J.-B. Pomet, Some explicit conditions for a control system to be feedback equivalent to extended Goursat normal form. In Proceedings of the IF AC Nonlinear Control Systems Design Symposium, (Tahoe City California, 1995). 2. R.Bryant, Some aspects of the local and global theory of Pfaffian systems. Ph.D. thesis (University of North Carolina, Chapel Hill, 1979). 3. R.Bryant, S-S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior Differential Systems. Mathematical Sciences Research Institute Publications. (Springer-Verlag, New York, 1991).

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4. M.Canadas-Pinedo and C.Ruiz, Pfaffian systems with derived length one. The class of flag systems. Transactions of the American Mathematical Society, 353, 1755-1766, (2001). 5. E.Cartan. Sur I'equivalence absolue de certains systemes d'equations differentielles et sur certaines families de courbes. Bulletin de la Societe Mathematique de France, 42, 12-48. (Euvres completes, Part. II, Vol. 2, (Gauthiers-Villars, Paris, 1914). 6. M.Cheaito and P.Mormul, Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 7 and 8. ESAIM Control, Optimisation, and Calculus of Variations, 4, 137-158 (1999). 7. G.Darboux, Sur le probleme de Pfaff. Bulletin des Sciences mathematiques, 2(6), 14-36, 49-68 (1882). 8. F.Engel, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften Mathematisch-Physikalische Klasse, Leipzig, 41,42, 157-176; 192-207 (1889), (1890). 9. M.Fliess, J.Levine, P.Martin, and P. Rouchon, Sur les systemes non lineaires differentiellement plats. Comptes Rendus de I'Academie des Sciences, 315, 619-624 (1992). 10. M.Fliess, J.Levine, P.Martin, and P.Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 6 1 , 6, 1327-1361 (1995). 11. G.Frobenius, Uber das Pfaff'sche problem. Journal fiir die reine und angewandte Mathematik, 82, 230-315 (1877). 12. R.Gardner and W.Shadwick, The GS algorithm for exact linearization to Brunovsky normal form. IEEE Transactions on Automatic Control, 37, 2, 224-230 (1992). 13. A.Giaro, A.Kumpera, and C.Ruiz, Sur la lecture correcte d'un resultat d'Elie Cartan. Comptes Rendus de I'Academie des Sciences de Paris, 287, 241-244 (1978). 14. E.Goursat, Lecons sur le probleme de Pfaff, ( Hermann, Paris, 1923). 15. D. Hilbert, Uber den Begriff der Klasse von Differentialgleichungen. Mathematische Annalen, 73, 95-108 (1912). 16. B. Jakubczyk, Invariants of dynamic feedback and free systems. In Proceedings of the European Control Conference, Groningen, (1993). 17. A.Kumpera and C.Ruiz, Sur I'equivalence locale des systemes de Pfaff en drapeau. In F. Gherardelli, editor, Monge-Ampere equations and related topics, (Instituto Nazionale di Alta Matematica Francesco Severi, Rome, 201-247, 1982). 18. P.Libermann, Sur le probleme d'equivalence des systemes de Pfaff non

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completement integrables. Publications Paris VII, 3, 73-110 (1977). 19. V. Lychagin, Local classification of non-linear first order partial differential equations. Russian Mathematical Surveys, 30, 1, 105-175 (1975). 20. P.Martin and P.Rouchon, Feedback linearization and driftless systems. Mathematics of Control, Signals, and Systems, 7, 235-254 (1994). 21. R.Montgomery and M.Zhitomirski Geometric approach to Goursat flags. (Preprint, University of California Santa Cruz 1999). 22. P.Mormul, Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 9. (Preprint, Institute of Mathematics Polish Academy of Sciences, 1997). 23. P.Mormul, Goursat flags: classification of codimension-one singularities. (Preprint, Warsaw University, 1999). 24. R.Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Mathematics of Control, Signals, and Systems, 7, 58-75 (1994). 25. P.Olver, Equivalence, Invariants, and Symmetry. (Cambridge University Press, 1995). 26. W.Pasillas-Lepine and W.Respondek, On the geometry of control systems equivalent to canonical contact systems: regular points,singular points, and flatness, in Proceedings of the 39th IEEE Conference on Decision and Control, 5151-5156 (Sydney, Australia, 2000). 27. W.Pasillas-Lepine and W.Respondek, Contact systems and corank one involutive subdistributions. To appear in Acta Applicand&Mathematicas (2001). 28. W.Pasillas-Lepine and W.Respondek, On the geometry of Goursat structures. ESAIM Control, Optimisation, and Calculus of Variations, 6, 119-181 (2001). 29. J-F. Pfaff, Methodus generalis, aequationes differentiarum partialum, nee non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi. Abhandlungen der KoniglichPreufiischen Akademie der Wissenschaften zu Berlin, Mathematische Klasse, 76-136 (1814-1815). 30. J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization. In B. Jakubczyk, W. Respondek, and T. Rzezuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, 32, 319-339. (Banach Center Publications, Warszawa, 1995). 31. D.Tilbury and S. Sastry, The multi-steering n-trailer system: A case study of Goursat normal forms and prolongations. International Journal of Robust and Nonlinear Control, 5, 4, 343-364 (1995). 32. A.Vinogradov, I.Krasil'shchik, and V.Lychagin, Geometry of Jet Spaces

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and Nonlinear Partial Differential Equations. (Gordon and Breach, New York, 1986). 33. E. von Weber, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften Mathematisch-Physikalische Klasse, Leipzig, 50, 207-229 (1898). 34. M.Zhitomirski, Normal forms of germs of distributions with a fixed segment of growth vector. Leningrad Mathematics Journal, (English translation), (2), 1043-1065 (1991).

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THE BRACHISTOCHRONE PROBLEM A N D MODERN CONTROL THEORY HECTOR J. SUSSMANN Department of Mathematics Rutgers University Hill center, Busch Campus Piscataway, NJ 08854, USA E-mail: [email protected]. edu URL: http://www.math.rutgers.edu/~ sussmann JAN C. WILLEMS Department of Mathematics University of Groningen P.O. Box 800, 9700 AV Groningen, The Netherlands URL: http://wvjw.math.rug.nI/~ willems Dedicated to Velimir Jurdjevic on his 60"1 birthday. 1

Introduction

The purpose of this paper is to show that modern control theory, both in the form of the "classical" ideas developed in the 1950s and 1960s, and in that of later, more recent methods such as the "nonsmooth," "very nonsmooth" and "differential-geometric" approaches, provides the best and mathematically most natural setting to do justice to Johann Bernoulli's famous 1696 "brachistochrone problem." (For the classical theory, especially the smooth version of the Pontryagin Maximum Principle, see, e.g., Pontryagin et al. 31 , Lee and Markus 2 6 , Berkovitz 3 ; for the nonsmooth approach, and the version of the Maximum Principle for locally Lipschitz vector fields, cf., e.g., Clarke 12 '', Clarke et al. 14 ; for very nonsmooth versions of the Maximum Principle, see Sussmann 35-36,39,4o,4i,42,43,44. for t n e differential-geometric approach, cf., e.g., Jurdjevic 24 , Isidori 19 , Nijmeijer and van der Schaft 3 0 , Jakubczyk and Respondek 2 0 , Sussmann 37 - 40 .) We will make our case in favor of modern control theory in two main ways. First, we will look at four approaches to the brachistochrone problem, presenting them in chronological order, and comparing them. Second, we will look at several "variations on the theme of the brachistochrone," that is, at several problems closely related to the one of Johann Bernoulli. The first of our two lines of inquiry will be pursued in sections 3, 4, 5, and 6, devoted, respectively, to

114

1. Johann Bernoulli's own solution based on an analogy with geometrical optics, 2. the solution based on the classical calculus of variations, 3. the optimal control method, and, finally, 4. the differential-geometric approach. We will show that each of the three transitions from one of the methods in the above list to the next one leads to real progress for the brachistochrone problem, by making it possible to derive stronger conclusions about it than those that were obtainable by means of previously existing techniques. Specifically, we will point out that l.a the differential equation derived by Johann Bernoulli has spurious solutions (i.e., solutions other than the cycloids), as was first noticed by Taylor in 1715, but l.b the application of the Euler-Lagrange condition of the calculus of variations eliminates these solutions and leaves only the cycloids, and that 2.a with the calculus of variations approach, the existence of solutions is a delicate problem, due to the non-coercivity of the Lagrangian, but 2.b the optimal control control method renders this question trivial, reducing the proof of existence to a straightforward application of the Ascoli-Arzela theorem, and, moreover, 3.a the usual calculus of variations formulations require that one postulate that the optimal curve is the graph of a function x >-> y(x), but 3.b with the optimal control control method this postulate becomes a provable conclusion.

115

These observations show that the first two of the three transitions brought improvements to the analysis of Johann Bernoulli's problem. But the core of our argument is 6, where we look at the third transition and the advent of differential-geometric control theory. We will argue that, on the one hand 4.a previous methods require that we assume extra knowledge of physics (in the form of the law of conservation of energy) in order even to be able to translate Johann's problem into mathematics, and 5.a with those methods an anomalous situation appears to arise when one looks for a state space formulation of the problem and asks for its "Hamilton principal function" V (that is, the value function regarded as a function of both the initial and terminal states), since V ought to be a function of an even number v of variables, but for the brachistochrone problem the best choice of v seems to be five; but on the other hand 4.b in the differential-geometric framework one does not need to bring in conservation of energy in order to formulate the problem; instead, one writes the equations as a control system in ffi4, and then computes the accessibility distribution, which turns out to be three-dimensional at each point; this shows that a nontrivial conserved quantity exists locally, and direct computation shows that it exists globally and is equal to the energy, and 5.b the anomaly described in 5.a disappears, and the principal function V turns out to be defined on E 4 x K4 (although V(q, q') < +00 if and only if q and q' belong to the same leaf of the foliation of R4 by the level submanifolds of the energy function); the method provides a full explanation for the apparent anomaly, by showing that the full optimal synthesis requires impulse controls that lead to instantaneous jumps. The discussion in 6 will also show that the differential-geometric approach brings into the problem interesting connections with other mathematical ideas. To begin with, the analysis leads naturally to a minimum time optimal control problem with a four-dimensional state space and an unbounded scalar control entering linearly into the dynamics. Moreover, this problem is quite degenerate, because its accessibility distribution happens to be three-dimensional, as

116

pointed out above. Since the problem is affine in the control but not linear, complete controllability on each leaf is in principle problematic. The accessibility Lie algebra turns out to be the "diamond Lie algebra," a well known four-dimensional solvable Lie algebra that occurs, for example, in the study of the quantized harmonic oscillator. To establish controllability on each leaf one needs additional arguments, and it turns out that the issue can be settled by computing the Jurdjevic-Kupka Lie saturate (cf. Bonnard et al. 8 ' 9 , Jurdjevic-Kupka 21>22; Jurdjevic 24 ) of the family of system vector fields. The minimum time problem then makes sense in principle for every pair of points belonging to the same leaf. But, since the control is unbounded, it is not at all obvious that an optimal control joining A to B exists for every pair (^4, B) of points belonging to a given leaf L. An application of the Maximum Principle yields a vector field Z tangent to the leaves such that all optimal arcs are integral curves of Z. Since, in general, two arbitrarily chosen points A, B of a leaf L do not lie on the same integral curve of Z, it follows that for general pairs (A, B) € L x L an optimal control joining A to B does not exist. On the other hand, when a control system is completely controllable and a suitable properness condition is satisfied a , then most mathematicians would agree that optimal controls ought to exist, and if they do not then this can only happen because we have not made a good choice of our space U of admissible controls, so U has to be enlarged by adding suitably defined "generalized controls." (For a trivial example, consider the problem of moving from A to B in minimum time with the dynamics x = u, u arbitrary, so in particular no bound is prescribed for the velocity u. Clearly, the "minimum time," is 0, and to attain it one needs to allow instantaneous jumps, which means that the class of ordinary controls has to be enlarged to allow for Delta functions. We will show in 6 that something like this happens for the brachistochrone.) In our case, the true complete optimal synthesis of the problem turns out to require impulse controls, a fact known to be common in problems with "cheap" unbounded controls, and studied in other (linear quadratic) contexts by Jurdjevic and Kupka, cf. Jurdjevic-Kupka 23 , Kupka 2 5 . Our second group of arguments in favor of modern control is presented in 7, and consists of "five modern variations on the theme of the brachistochrone." Following a suggestion made by M. Hestenes in 1956, we "speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times" (Hestenes 18 , p. 64), and we "See Definition 6.3 for a precise statement of the properness condition, and Proposition 6.4 for the proof that the condition is satisfied for the brachistochrone problem.

117

extend Hestenes' idea by speculating not only on "what type of . . . problem the . . . brothers might have formulated" but also on what type of answer they might have liked for such problems. We deal with five questions. The first one is Hestenes' own "variation of the brachistochrone problem," which turns out to be a problem solvable by optimal control but not by classical calculus of variations techniques. The second "variation" is the "reflected brachistochrone," a very natural extension of Johann's problem, in which the state space is the whole plane rather than a half plane. This problem is no longer solvable by means of traditional smooth or nonsmooth optimal control methods—because it is a minimum time optimal control problem with a non-Lipschitzian right-hand side—but turns out to lend itself to the "very nonsmooth" approach, which yields a solution with a simple geometric interpretation that, in our view, the Bernoulli brothers would have liked. The third "variation" is the brachistochrone problem with friction, which corresponds, mathematically, to "unfolding" the original Johann Bernoulli question by embedding it in a one-parameter family of problems, the parameter being, of course, the friction coefficient p. These problems turns out to be nondegenerate, in the sense that, for example, their accessibility distribution has rank four at each point. Consequently, their time-optimal controls behave qualitatively like the time-optimal controls for generic control-affine systems in R4 with scalar unbounded controls. In particular, through every point there pass a one-parameter family of optimal curves, since the optimal control is given in feedback form as a function of the state and of an additional scalar parameter that stays constant along each trajectory. The analysis clarifies the way in which the original problem of Johann Bernoulli is degenerate: the accessibility Lie algebra is eight-dimensional for p ^ 0 but four-dimensional for p = 0; the accessibility distribution has rank four everywhere if p ^ 0 but has rank three everywhere if p = 0; the optimal "feedback" control is a function of the state and an extra parameter, but this parameter only occurs in a term that is multiplied by p, so that when p = 0 the optimal control is a function of the state only. The fourth "variation" deals with the derivation of Snell's law of refraction from general necessary conditions for an optimum. Since Snell's law plays a crucial role in Johann's solution of the brachistochrone problem, we speculate that a natural question for him to have asked "if he had lived in modern times" is whether the general theories about minimizing curves that were created in the three centuries after Johann's work actually cover the Snell law case. Our answer is negative if only the classical calculus of variations and pre-1990 smooth and nonsmooth optimal control are considered, but becomes positive

118

if we add to the list our recent very nonsmooth methods To conclude the paper, we take the liberty of including a fifth "variation" that we find amusing but does not really have much to do with control theory. In this "variation"—which is a digression on a topic of great importance and related to geometry, to the brachistochrone, and to Johann Bernoulli—we "speculate" that Johann Bernoulli would have found it very interesting to know b that with his work on the brachistochrone he had been extremely close to solving one of the most important open problems in Geometry, namely, that of the independence of Euclid's fifth postulate from the other ones. (The problem had intrigued mathematicians since ancient times—cf., e.g., Bonola 10 —and was very much in the forefront of research in geometry in Johann's times, as shown by the publication of Saccheri's book 32 in 1733. A definitive solution was found by Beltrami in 1868, cf. Beltrami 2 and Stillwell 33 .) Remark 1.1. An earlier, related version of the argument presented here appeared in 3 8 . That paper also dealt with the evolution from the brachistochrone to the calculus of variations and optimal control, but focused on a different set of issues, especially on the search for the correct Hamiltonian formulation of the necessary condition for an optimum and how opportunities were missed before the "control Hamiltonian" was finally discovered in the 1950s. Q 2

Johann Bernoulli and the brachistochrone problem

In the June 1696 issue of the journal Acta Eruditorum, Johann Bernoulli—a Swiss-born professor of mathematics at the University of Groningen, in the Netherlands0—challenged mathematicians to solve the "problema novum" of finding the curve that—following Leibniz' suggestion—he named the "brachistochrone," from the Greek /3paxicroq: shortest, and xP^vcx;: time d . The journal's May 1697 issue contains articles on this problem by six of the most renowned mathematicians of the time: Johann Bernoulli, Johann's elder brother Jakob, Leibniz, Newton, the marquis de l'Hopital, and Tschirnhaus. Leibniz called Johann Bernoulli's question a "splendid" problem. The fact that he and five other thinkers of such unparalleled distinction chose to * Although he probably would not have thought of asking the question. c T h e Bernoulli family was originally from Antwerp in Flanders, where they lived until 1583. Like many other Flemish Protestants, they fled Flanders to escape religious oppression by the Spanish rulers. They spent some time in Frankfurt, and finally settled in Basel, Switzerland, early in the seventeenth century. d So the brachistochrone is the solution of a minimum time problem, i.e., of a genuine optimal control problem in the modern sense of the term!

119 become actively engaged in the search for its solution shows that at the time of its formulation the problem was perceived by the leaders of the mathematical community as a major challenge, lying at the very frontiers of their research. It would not have been easy at that point to predict the future impact on science of Johann Bernoulli's challenge. Yet, the six famous authors of the 1697 papers must have had their own reasons to sense that an exceptionally promising line of inquiry was being opened, and the developments of the 300 years that followed have lent ample support to their intuition. Today, it is clear that the events of 1696-97 represented a critical turning point in the history of mathematics, and it is widely agreed that with them an important new field, later to be called the "calculus of variations," was born. R e m a r k 2 . 1 . The words "calculus of variations" were first used by Euler in a 1760 paper, cf. 1 5 . Euler had received, in 1755, a letter from a 19-year-old mathematician from Turin, called Ludovico de la Grange Tournier, where a new method was proposed for the study of variational problems, by using what we would now call "variations" (cf. Goldstine ie, p. 110.). This was quite different from the approach Euler had been using until then, and presented in his 1744 book entitled Methodus inveniendi maximi minimeve proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti, that is "A method for discovering curved lines having a maximum or minimum property or the solution of the isoperimetric problem taken in its widest sense." Euler's earlier method was based on a time-discretization followed by a passage to the limit?, as the size h of the time intervals goes to 0. Euler was so enthusiastic about the new idea of the unknown youngster from Turin that he dropped his own method, adopted that of Lagrange instead, and renamed the subject the calculus of variations. In the summary to his first paper using variations, Euler says "Even though the author of this [Euler] had meditated a long time and had revealed to friends his desire yet the glory of first discovery was reserved to the very penetrating geometer of Turin LA GRANGE, who having used analysis alone, has clearly attained the same solution which the author had deduced by geometrical considerations." £> T h e newborn subject was destined t o become a major area of m a t h e matics. Moreover, its implications would soon extend t o physics, whose very foundations would b e shown in t h e 18th century—by Euler, Maupertuis, Lagrange, and others—to depend on principles formulated in t h e language of the calculus of variations. So it is not surprising t h a t t h e story of t h e brachistochrone is told in e

Euler himself talked about "letting h be infinitesimal."

120

many books, and that the importance of this curve is stressed in most histories of mathematics. The brachistochrone has received honors not commonly bestowed on mathematical curves. For example, it is depicted in a stainedglass window of the Academy building (the main venue of the university) in Groningen. Moreover, the brachistochrone is probably the only mathematical curve that has been deemed worthy of a monument in its honor. In 1994, on the occasion of the 375th anniversary of its foundation, the University of Groningen decided to honor in various ways the most famous former members of its faculty. In the case of Johann Bernoulli, who had been a professor from 1695 to 1705, a decision was made to honor him by erecting a monument to the brachistochrone, one of the most significant discoveries—and one of which he was particularly proud—that he made during his Groningen period. The unveiling of the brachistochrone monument, located in the Zernike complex of the University, took place in 1996, coinciding with the 300th anniversary of the publication of Johann's challenge in Acta Eruditorum. 2.1

Johann Bernoulli's challenge

The June 1696 challenge took the form of an "Invitation to all Mathematicians to solve a new problem": If in a vertical plane two points A and B are given, then it is required to specify the orbit AMB of the movable point M, along which it, starting from A, and under the influence of its own weight, arrives at B in the shortest possible time. So that those who are keen of such matters wiJJ be tempted to solve this problem, is it good to know that it is not, as it may seem, purely speculative and without practical use. Rather it even appears, and this may be hard to believe, that it is very useful also for other branches of science than mechanics. In order to avoid a hasty conclusion, it should be remarked that the straight line is certainly the line of shortest distance between A and B, but it is not the one which is traveled in the shortest time. However, the curve AMB—which I shall disclose if by the end of this year nobody else has found it—is very well known among geometers. Later, Johann Bernoulli changed his original plan to disclose the solution by the end of 1696 and, following a suggestion by Leibniz, extended the deadline until Easter 1697.

121

Responses came from the best mathematical minds of the time. In addition to Johann's own solution, there was one by Leibniz, who had communicated it to Johann in a letter dated June 16, 1696; another one by Jakob; one by Tschirnhaus; one by l'Hopital and, finally, one by Newton. Newton's solution was presented to the Royal Society on February 24, 1697 and published, anonymously and without proof, in the Philosophical Transactions. (The identity of the anonymous author was clear to Johann Bernoulli, since he thought that you can know ex ungue leonem, i.e., that you can know "the lion from its claws.") The May 1697 issue of Acta Eruditorum contains articles with the solutions by Johann 5 and Jakob 6 , as well as contributions by Tschirnhaus and l'Hopital, a short note by Leibniz, remarking that he would not reproduce his own solution, since it was similar to that of the Bernoullis, and Newton's anonymous paper, reprinted from the Philosophical Transactions. 2.2

The cycloid's double role as the brachistochrone and the tautochrone

As is apparent from Johann Bernoulli's announcement, he was under the impression that the problem was new. However, Leibniz knew better: Galileo in his book on the Two New Sciences in 1638 formulated the brachistochrone problem and even suggested that the solution was a circle. (Galileo had showed—correctly—that an arc of a circle always did better than a straight line—except, of course, when B lies on the same vertical line as A—but he then wrongly concluded from this that the circle is optimal.) Johann Bernoulli considered the fact that Galileo had been wrong on two counts, in thinking that the catenary was a parabola, and that the brachistochrone was a circle, as definitive evidence of the superiority of differential calculus (or the Nova Methodus, as he called this new set of techniques, following Leibniz). He was thrilled by his discovery that the brachistochrone was a cycloid. This curve had been introduced by Galileo who gave it its name: related to the circle. Huygens had discovered a remarkable fact about the cycloid: it is the only curve with the property that, when a body oscillates by falling under its own weight while being guided by this curve, then the period of the oscillations is independent of the initial point where the body is releasedJ Therefore Huygens called this curve, the cycloid, the tautochrone (from TOLVTCX;: equal, and xpovcx;: time). Johann Bernoulli was amazed and somewhat puzzled, it seems, by the fact that the same curve had these two remarkable properties related to the 'Contrary to what Galileo thought, the circle has this property only approximately: the period of oscillation of a pendulum is a function of its amplitude.

122

time traveled on it by a body falling under its own weight: Before I end I must voice once more the admiration that I feel for the unexpected identity of Huygens' tautochrone and my brachistochrone. I consider it especially remarkable that that this coincidence can take place only under the hypothesis of Galilei [HJS
2.3

Why was the brachistochrone so important in 1696-97?

Perhaps the crucial fact about Johann Bernoulli's question is that he is telling us very clearly that the solution is a curve that everybody knows, and yet there appears to be no way to guess which curve it is by means of some intuitive geometric argument. This situation should be compared with that arising in another famous and much older question about curve minimization, the isoperimetric problem. In this problem, we are asked to find the simple closed rectifiable curve that encloses the largest possible area among all simple closed rectifiable curves of a given length, and the answer also turns out to be a curve that is certainly "very well known among geometers," namely, a circle. But in this case it is fairly easy to guess directly that the answer is a circle, even though it is a delicate matter to give a rigorous proof. For example, it suffices to observe that the question is rotationally symmetric, so the answer ought to be a rotationally symmetric curve, and the only such curve is the circle. Although this argument does not amount to a completely rigorous proof9, it is undoubtedly rather convincing, and makes it quite obvious that the solution must be a circle. Therefore, if Johann Bernoulli had challenged "the best mathematical minds" of his time to solve the isoperimetric problem, it would not have made s

T h e argument assumes the existence of a solution, as well as its uniqueness up to translations. Indeed, it might conceivably happen that the solution is not unique, and in that case the only conclusion one can draw from rotational symmetry is that every rotation will map a solution 5 to a solution S', which may be different from S.

123

sense for him to say, as he did in his brachistochrone challenge, that "the curve—which I shall disclose if by the end of this year nobody else has found it—is very well known among geometers," since everybo,y would have guessed immediately that the mysterious curve was the circle. In the case of the brachistochrone, there is no way to guess h. The problem was posed and solved at a time when the Calculus was being invented, and cannot be solved without using Calculus. So it provides an impressive illustration of the power of the Calculus to solve problems that cannot be solved in any other way, and to discover truths about the world that can be stated and understood in "well known" terms ', but can only be arrived at by using this new mathematical tool. 3

The standard formulation and Johann Bernoulli's solution

We now discuss the "standard formulation" of the brachistochrone problem in modern mathematical language, as it is usually found in modern books, and Johann Bernoulli's ingenious solution based on an analogy with geometrical optics. In our view, the standard formulation is questionable. We will explain our objections in 6 below, where we will make a case for a different approach, based on differential-geometric control. But first it is necessary to present the classical point of view in detail. The key fact is that the "movable point M" is supposed to move "under the influence of its own weight," which presumably means "under the influence of its own weight and nothing else other than whatever force is needed to keep M on the given curve." This means, in particular, that energy is conserved. 3.1

Formulation of the brachistochrone problem

We choose x and y axes in the plane with the y axis pointing downwards, use (a, a) and (b, j3) to denote, respectively the coordinates of the endpoints A fe Even now, 300 years later, we are not aware of any simple geometric argument that might lead to the cycloid without using Calculus. ' T h i s is an important point. Compare the brachistochrone, for example, with Newton's 1685 work on the shape of a body with minimal drag. In this case, the solution also requires the Calculus, but once we have solved the problem all we find is a curve defined by some strange formula. This curve was not "well known to geometers," and would not have had any particular significance to mathematicians, who would only retain from going through Newton's solution the fact that the solution is "some curve." In the case of the brachistochrone, one does not need the Calculus to explain what the solution is, since cycloids were already well understood curves, but one does need it to discover that the solution is a cycloid.

124

and B of our curve, and fix a number B e l , thought of as the total energy of the particle. The condition characterizing the feasible paths [0,T] B t H- ^(t) = (*(*)»!/(*)) e K2

(3.3.1)

will then be—using m, g to denote, respectively, the particle's mass and the gravitational acceleration—that the sum of the kinetic energy ^{xty)2 +y(t)2) and the potential energy —gmy(t) is equal to E. We can certainly choose units so that m = 1. The condition then becomes l

-(i(t)2

+ y{t)2)-gy{t)

= E.

So we formally define an {A, B, E)-feasible trajectory (or (A, B, E)-feasible path) to be a map (3.3.1), defined on some interval [0, T], such that (i) t(0) = A, £(T) = B, (ii) £ is absolutely continuous, (iii) \{x{t)2 + y(t)2) = E + gy{t) for almost all t G [0,T]. Notice that Condition (iii) implies in particular that the derivatives x and y are bounded, so a feasible path is in fact Lipschitz continuous. Condition (i) states that the path £ must start at A and end at B. The number E + ga is the initial kinetic energy of the body. (Often, only the case E + ga = 0 is considered, corresponding to the body being dropped at time 0 with zero initial speed.) Condition (ii) is of course a modern technicality that would have made no sense to Johann Bernoulli, who was far from imagining that there could exist curves that cannot be recovered by integrating their derivative. Condition (iii) reflects conservation of energy. (The law that the kinetic energy of a body which has fallen from a height h is increased by an amount proportional to h was due to Galileo, and was well known in Bernoulli's time.) An (A, B, .E)-feasible path £ : [0, T] -> E 2 is said to be optimal if there exists no (A, B, £)-feasible path f: [0, f ] -> E 2 for which f < T. An (A, B, E)brachistochrone is either (a) an optimal (A, B, E)-feasib\e path, or (b) the geometric locus in ffi2 of such a path, that is, a subset B of E 2 such that B = {(x,y) G E 2 : there exists t G [0,T*], such that (x,y) = £*(*)} , for some optimal (A, B, .E)-ieasible path £* : [0, T*] i-> E 2 . A brachistochrone is an object which is an (A, i?,.E)-brachistochrone for some A,B,E.

125

Obviously, a feasible path" (3.3.1) must be entirely contained in the closed half-plane H

+(v) = {(x,y) : y > -n] ,

where

n= —,

(3.3.2)

since, for y(t) to be < — n, the kinetic energy would have to be negative, which is of course impossible. It is clear that the solution cannot always be a straight line, as Johann Bernoulli rightly points out in his "Invitation." For example, consider the extreme case when E = 0 and a = ct = (3 = Qj£b. (That is, when the endpoints of the curve are at the same height but do not coincide, and the initial kinetic energy vanishes.) We will show that in this situation there is a feasible path that goes from A to B in finite time. Since the straight-line segment S from A to B is horizontal, (iii) implies that the speed of motion along S vanishes, and then it will follow that S cannot be an optimal path, because the motion along S takes infinite time. To construct a feasible path from A to B we follow a circle C with center on the x axis. The circle can be parametrized by x, taking values in the interval [0, b]. The speed v of motion along C will be proportional to ^/y, and hence bounded away from 0 as long as x is bounded away from 0 and b\. Near A, we can also use y as a parameter, and then y \, 0 as x ], 0, and v behaves like -^ and also like y/y, so ^ ~ -j=, showing that t stays finite as x and y approach 0, since the function y H-> -4= is integrable near y = 0. A similar argument shows that t stays finite near B. So C joins A to B in finite time. 0 3.2

Johann Bernoulli's

solution

It turns out that the brachistochrone is a cycloid. When E = 0, it is the curve described by a point P in a circle that rolls without slipping on on the x axis, in such a way that P passes through A and then through B, without hitting the x axis in between. (It is easy to see that this defines the cycloid uniquely.) Johann Bernoulli's ingenious derivation of the brachistochrone has been the subject of numerous accounts, but since this event plays a crucial role in our own tale, we will briefly tell the story again. Johann Bernoulli based his derivation on Fermat's minimum time principle. If we imagine for a moment that instead of dealing with the motion of a moving body we are dealing with a light ray, Condition (iii) above gives us a °FVom now on, as long as A, B and E are fixed, we just talk about "feasible paths" and "brachistochrones," without explicitly specifying A, B and E.

126

formula for the "speed of light" c(x,y) as a function of the position (x,y): c(xiy)

= ^/2E + 2gy.

We now change coordinates so that E = 0. (This amounts to moving the x axis upward to a height rj.) Then r) is now equal to 0, so our feasible paths live in the half plane H+ = H+(0). Also, let us rescale—or, if the reader so prefers, "change our choice of physical units"—so that 2g = 1. Then our problem is exactly equivalent to that of determining the light rays—i.e., the minimum-time paths—in a plane medium where the speed of light c varies continuously as a function of position according to the formula c[x,y) = y/y. It is then reasonable to expect that, if we use a suitable discretization of our problem, then the optimal paths of the discretized problem will approach those of the original problem as the discretization parameter 5 goes to 0. Johann Bernoulli did this as follows. Let us define ysk=kS

for

S > 0 and it = 0 , 1 , . . . ,

and then divide the half-plane H+ into horizontal strips Sskd={(x,y)eR2:yi
for

k = 0,1,...,

of height S. Johann's discretization is then obtained by replacing c by a constant csk in each strip Sk. The constants c\ are defined by

4 = yvi+i

for

each

k

•

He then computed the light rays for the original problem by taking the limit of the light rays for the discretized problem as 6 j . 0. The light rays of the discretized problem can be studied using Snell's law. Clearly, the paths will be straight-line segments within each individual strip, and all that needs to be done is to determine how these rays bend as they cross the boundary between two strips. The answer is provided by the laws of optics as developed by Snell, Descartes, Fermat, Leibniz and Huygens. Snell had observed that, if the speeds of light on the two sides Hi, Hi of a line L in the plane are different constants «i, i>2, and a light ray R traverses Hi and then enters H2 after being refracted at L, then the ratio fj^V" °^ ^ e s m e s of the incidence and refraction angles is a constant, independent of 61. (By definition, 9i is, for i = 1,2, the angle formed by the part Ri of R that lies in Hi with the line perpendicular to L. It is clear that the Ri are straight lines.) Fermat subsequently showed that this is precisely what happens when light

127

is assumed to follow a minimum-time path. The law relating the incidence angles to the velocities of propagation is due to Leibniz and Huygens, and implies the law of Snell. It says that sin 0i vi ——— = — sin0 2 v2

. or, equivalently,

sin 0i sin0 2 = , vi v2

,,,,>

v(3.3.3)

Johann Bernoulli used (3.3.3) to study the light rays of the discretized problem. If R is such a ray, and we let Rk be the part of R that lies in the strip S|, and use 8sk to denote the angle of Rk with a vertical line, then (3.3.3) enabled him to conclude that the quantity

sin0f

t1/in is a constant, since in each strip S% the speed of our light ray is t/yjfe+1. Passing to the limit as 5 \. 0, we conclude that the sine of the angle 0 between the tangent to the brachistochrone and the vertical axis must be proportional to ^/y. Since dx

sin 0 = sjdx

1

+ dy2

,

we find that dx2 dx + dy2 ~ 2

Vy

'

where v is a constant. Then dx2 4- dy2 _ 1 dx2 vy I.

e., 1 + y'(x)2

C 1 - — , where C = - .

So the curve described by expressing the y-coordinate of the brachistochrone as a function of its x-coordinate will satisfy the differential equation

y'(x) = J^f^, V y(x) for some constant C.

(3.3.4)

128

The parametrized curves tp i-> (x((p),y(
C y(ip) = —(1-cos


0<
satisfy (3.3.4). It is easily seen that these equations specify the cycloid generated by a point P on a circle of diameter C that rolls without slipping on the horizontal axis, in such a way that P is at {XQ, 0) when ip = 0. Moreover, it is also easy to check that (*) given two points A and B in H+ there is exactly one curve passing through A and B and belonging to the family of curves (3.3.5). (The parameter count for (*) is obviously right: (3.3.5) describes a 2parameter family of curves; requiring that the curve pass through A fixes one of the parameters, and then asking that it pass through B fixes the other. Proving (*) requires a bit more work, but is still quite easy.) 4 4-1

Spurious solutions and the calculus of variations approach Remarks on Johann Bernoulli's

argument

The argument that we have presented is the one of Johann Bernoulli, and Equation (3.3.4) is the one that he wrote in his paper, followed by the statement "from which I conclude that the Brachistochrone is the ordinary Cycloid." (He actually wrote dy = dx./-^^, but he was using x for the vertical coordinate and y for the horizontal one. Cf. 34 , p. 394.) In present day notation, it is customary to adopt the convention that the symbol y/r stands for the nonnegative square root of r, but it is obvious that Johann Bernoulli did not have this in mind. What he meant was, clearly, what we would write today as

or, equivalently, y(x)(l + y'(x)2) = constant.

(4.4.2)

In particular, the solution curves should be allowed to have a negative slope, contrary to what a "modern" interpretation of (3.3.4) might suggest. But y' should stay continuous, so that a switching from a + to a — solution of (4.4.1) is not permitted.

129 Even with the more accurate rewriting (4.4.2), the differential equation derived by Johann Bernoulli also has spurious solutions, not given by (3.3.5). Indeed, for any y > 0, the constant function y(x) = y is a solution, corresponding to C = y. More generally, one can take an ordinary cycloid given by (3.3.5), follow it up to y>=7r—so that dy/dx=Q—then follow the constant solution y(x)=C for an arbitrary time T, and then continue with a cycloid given by (3.3.5). Such paths are, indeed, compatible with Huygens' law of refraction. Remark 4.1. The existence of the spurious solutions was pointed out by Brook Taylor (1685-1731) in 1715. (Taylor is known to mathematicians as the discoverer of "Taylor's expansion," the inventor of integration by parts, and the creator of the calculus of finite differences. Like many other mathematicians of his time, he was involved in a priority dispute with Johann Bernoulli, involving Taylor's 1708 solution of the problem of the center of oscillation, that went unpublished until 1714.) <£ 4-2

The calculus of variations approach

The spurious solutions, and all the other problems, such as the apparent arbitrariness of the requirement that y' be continuous, can be eliminated in a number of ways. For example, one can prove directly that the spurious trajectories are not optimal. Or one can use, as an alternative to Johann Bernoulli's method, the calculus of variations approach, based on writing the Euler-Lagrange equation. This equation gives a necessary condition for a function y* to minimize the integral

J= /

L(y(x),y'(x),x)dx

Ja

within the class y of all functions [a,b] B xt-^ y(x) e R that satisfy some appropriate technical conditions (for example, the requirement that y(-) is Lipschitz continuous) and are such that y(a) and y(b) have given values a, j3. Here, the "Lagrangian" 0 x R x [a, b] 3 {y,u,x) >->• L(y,u,x)

GR

is a given function, and Q is an open interval in R. Under suitable technical

130

conditions on L, yt, and the set y, if j/* is a minimizer then the condition

dx \ du X

V* (x) ,y'.(x),z))

=-Q- (v* (x), y', (x), x)

(4.4.3)

must be satisfied for almost all x. It is easy to see that the above result can be applied to the brachistochrone problem, provided that we postulateb that it suffices to consider curves in the x, y plane that are graphs of functions x t-> y(x) denned on the interval [a, b]. Then Constraint (iii) can be written (taking E = 0 and 2g = 1 as before) as dx2 +dy2 = y dt2 , which gives dt =

v/cte2 + dy2 — = L(y, y )dx.

where

L(y,u) = ^ ± Z .

(4.4.4)

Vy So Johann Bernoulli's problem becomes that of minimizing the integral J = /

L{y(x),y'{x))dx,

J a

subject to y(a) = a and y(b) = /?, where L is given by (4.4.4). This gives the equation l + y'(x)2+2y(x)y"(x)

= 0,

(4.4.5)

which is stronger than (4.4.2), since (4.4.2) is equivalent to y' + y' +2yy'y" = 0, i.e., to j/'(l + y' + 2yy") = 0, whose solutions are those of (4.4.5) plus the spurious solutions found earlier. It is easy to verify that the solutions of the Euler-Lagrange equation (4.4.5) are exactly the curves given by (3.3.5), without any extra spurious solutions, showing that, for the brachistochrone problem, the Euler-Lagrange method gives better results than Johann Bernoulli's approach. We now show that optimal control is even better. ''With optimal control, this "postulate" becomes a provable conclusion, as will be shown in 5 below.

131

5

The optimal control approach

Having shown that the Euler-Lagrange equation gives better results than Johann Bernoulli's method, we now look at another important step in the evolution of the theory of minimizing curves, namely, optimal control theory. We show that optimal control provides even better results in two ways: 1. With optimal control it is no longer necessary to postulate that the optimal curve is the graph of a function x i-» y(x). 2. In the optimal control setting, the problem of the existence of an optimum becomes trivial. 5.1

The brachistochrone as an optimal control problem

We can formulate Johann Bernoulli's question as an optimal control problem, in which the motion takes place in the x, y plane, and the dynamical behavior is given by x = u^/\y\,

y = vy/\y~\.

(5.5.1)

Here the control is a 2-dimensional vector (u, v) taking values in the set U = {(u,v):u2+v2

= 1}.

(5.5.2)

(Actually, for the "true" brachistochrone problem the motion is restricted to the upper half plane, so we could have written y/y rather than y/\y\. We choose to use the more general expression y\y\ because later, in the study of the reflected brachistochrone, we will want to work in the whole plane.) Let us use p, q, p0 to denote, respectively, the momentum variables conjugate to x and y, and the abnormal multiplier. Then the control theory Hamiltonian H(x,y,u,v,p,q,po,t) is given (using a = sgny) by the formula H(x,y,u,v,p,q,p0,t)

= (pu + qv)y/ay - p 0 .

The Pontryagin Maximum Principle then tells us that if a curve [0,T] 9 i .->£(*) = (*(*),»(*)) is optimal then there exist absolutely continuous functions t »->• p(t) and 11-> q{t), and a nonnegative constant po, such that, if we let u = x, v — y, and write p to denote the momentum vector (p, q), and \\p\\ to denote its Euclidean norm y/p2 + q2, then the Hamiltonian maximization conditions

• ( 0 -iwoii' "<,, = »

(5 5 3)

'-

132

as well as the "adjoint system" of differential equations

«„=„,

W-_*'*M±%M«)__-4ai 2

2 vow)

,

(5.5.4)

v<w)

are satisfied for almost all t. (Notice that \\plt)\\ ^ 0. Indeed, the Maximum Principle also gives the condition that H = 0. So ||p|| = 0 would imply p0 = 0, contradicting the nontriviality of the triple (p(t),q(t),p0).) If the constant p vanishes, then x = 0, so we get a vertical line. Otherwise, x is always ^ 0, showing that we can use x to parametrize our solution. Since y>{x)

=

!k ax

=

i = l = lt x u p

(5.5.5)

we have l+y'(x)2

= ^ -

(5.5.6)

and y

„{x)

=

l.*L p dx

i.. px

=

(s.5.7)

But (5.5.1) and (5.5.3) imply that a; =

*

Ibll

,

(5.5.8)

and then (5.5.4) and (5.5.7) yield »"(*) = -

2 ML & • 2

<5-5-9)

2t/P So 2yy" = - ^

= - ( 1 + y'2), and then l + y'2+2yy"

= 0,

(5.5.10)

which is exactly Equation (4.4.5). As before, this leads to the cycloids, with no "spurious solutions." Notice that this argument does not involve any discretization or any use of Snell's law of refraction. Notice also that in our control argument we have not postulated that the solution curves could be represented as graphs of functions y(x). We have proved it! This should be contrasted with the analysis based on the calculus of variations, where this fact had to be postulated.) This is one example showing that, for the brachistochrone problem, the optimal control method gives better results than the classical calculus of variations.

133

5.2

Optimality proofs and the existence problem

For a second example showing the advantages of the optimal control method over the classical calculus of variations for the specific case of the brachistochrone problem, we turn to the question of the rigorous proof of the optimality of Bernoulli's cycloids. Clearly, no argument based solely on necessary conditions for optimality will ever prove that a trajectory is optimal, because it could happen, for example, that there are no optimal trajectories, in which case the statement that "every optimal trajectory is a cycloid" would be vacuously true. If we really want to prove the optimality of Bernoulli's cycloids, an extra step is needed. To give a rigorous proof of the optimality of the cycloids, valid for all choices of A and B in the closed half-plane H+ = {x,y) : y > 0}, one can proceed in two ways. One possibility (cf. Bliss 7 ) is to use Hamilton-Jacobi theory. Another, much simpler, approach, is to prove first the existence of an optimal trajectory. Indeed, once this existence result is known for all A, B 6 H+, then it is clear that any optimal trajectory £» : [0, T] t-> H+ going from A to B must be such that its restriction £, \[a, /?] to every subinterval [a, /?] of [0, T] is also optimal for the problem P(a, /?) with endpoints £»(«), £*(/?). (This is a rather obvious point, known in optimal control theory as the "principle of optimality," and noticed in the case of the brachistochrone by Jakob Bernoulli.) If one assumes that £» (a) and £* (/?) are interior points of H+ for 0 < a < /3 < T, then the necessary conditions for optimality give a unique candidate for the solution of P(a,/3), namely, the unique curve in the family given by (3.3.5) that goes through £*(a) and £»(/?). So £* is such that ^»[[a,/?] is given by (3.3.5) whenever 0 < a < /? < T, and this easily implies that £* itself is a cycloid given by (3.3.5). To complete the argument, one has to exclude the possibility of a solution of the problem with endpoints A, B which intersects the x axis at some other point. This can be done by a simple qualitative argument, using the following lemma. L e m m a 5.1. Suppose L is a horizontal line contained in the interior of the upper half-plane H+. Let P, Q be distinct points of L. Then the straight-line segment joining P to Q is strictly faster than any trajectory C, from P to Q such that £ lies entirely in the lower closed half-plane determined by L and at least one point of C, lies strictly below L. Proof. Let T^, T^ be the corresponding times. Let y be the y coordinate of the points of L, so y > 0. Write P = (p,y), Q = (q,y). Let £ : [0,T] >-> H+

134 be the horizontal segment from P to Q. Then Te = Now let [a,b] 3 t i-> ((t) = (x(t),y(t)) in our statement. Then

\q-p\

Vy ' G H+ be a trajectory from P to Q as

m< warn = Vv(d
< ( 6 - o ) v ^ = T c V».

JO

Therefore T, > i| g^ ~- ^Pil = r € , and our proof is complete.

0

d

The following is an immediate consequence of the lemma. Corollary 5.2. Lei A, B be points of H+ such that A^ B, and assume that £ : [a, b] i-+ if+ is h'me optimal. Then £(t) is an interior point of H+ whenever a •

135

So everything hinges upon the existence result. It turns out that this is not at all trivial if the brachistochrone problem is reformulated in terms of the classical calculus of variations 0 , because (a) the Lagrangian L given by L(y, u) =

t^

has a singularity at y = 0,

and, even more importantly, (b L is not sufficiently coercive for the usual existence theorems to apply. (The precise meaning of (b) is that, as a function of u, L just grows like a constant times |u|, whereas for the usual existence theorems to apply L would have to grow like a constant times |u| r for some r such that r > 1. More sophisticated existence theorems for classical calculus of variations problems that include the brachistochrone are stated in Cesari n , 14.3 and 14.4, but these conditions are not at all simple, and in addition they depend on other results by Cesari that were to appear in a book that was never published.) In the control theory setting, however, the existence question is easily settled by a completely trivial application of the Ascoli-Arzela theorem: given A, B £ H+, let T be the infimum of the times of all trajectories going from A to B; let {£,-} be a sequence of trajectories from A to B, defined on intervals [0, Tj], such that Tj I T. Then the £y are all contained in a fixed compact subset of H+, as can easily be shown by applying Gronwall's inequality and using the fact that the right-hand side of (5.5.1) has less than linear growth as a function of x and y. Then (5.5.1) implies that the derivatives £, are uniformly bounded as well. So we can use the Ascoli-Arzela theorem to extract a subsequence that converges uniformly to a curve £ : [0,T] \-¥ H+. In principle, £ is just a trajectory of the "convexified" system, in which the evolution equations are those of (5.5.1) but the control constraint is u2 + v2 < 1 rather than u2 + v2 = 1. However, it is easy to see that £ is a true trajectory of the nonconvexified system because, if the inequality u(t)2 + v(t)2 < 1 was strict on a set of positive measure, then it would be possible to find an even faster trajectory from A to B by just reparametrizing £. So we see, once again, that in the control setting a question that is hard from the point of view of the calculus of variations can become easy or even trivial. The key point in the above argument is that there is at least one trajectory going from A to B in finite time, as can easily be verified. (If one of the points J4, B belongs to dH+, and A ^ B, then the argument would not work if the speed of light was y rather than ^/y, for in that case there would be no path joining A and B in finite time.) c

We thank F. H. Clarke for bringing this point to our attention.

136

6 6.1

The differential-geometric connection Is the standard formulation of the brachistochrone problem satisfactory?

The mathematical formulation of the brachistochrone problem, as presented in 3.1, is not completely natural, because it takes it for granted that we know that energy is conserved. More precisely, "energy conservation" is "assumed" in at least two conceptually different ways. First, the fact of energy conservation is a prerequisite for even stating the problem. This is clearly not intellectually satisfactory, because it would be much better to state the problem first, and then let the mathematical analysis discover energy conservation for us, after which we could use this new fact to find the solution. Second, the problem of "finding the minimum time curve from A to B" does not make sense for general A and B, unless the value of the energy E is specified. The problem usually studied in textbooks is another one, namely, that of "finding the minimum-time curve from A to B subject to an extra constraint E = E, for a given E." If this constraint is not given, then we really have, for each given pair (^4, B) of endpoints, a one-parameter family of problems, one for each value of E. (Alternatively, we may want to look at the unconstrained problem, in which we are also minimizing over all values of E. But this is clearly not what Johann Bernoulli or anyone else who studied the problem had in mind, and in addition is easy to solve: the infimum of the times is zero, and there is no minimizer.)

6.2

Hamilton's principal function

At this point, we digress to discuss an important idea of Hamilton, namely, the "principal" or "characteristic" function. This idea was not truly appreciated for more than century, until optimal control theory emerged to provide the right setting that would do it justice. So we will only describe it here in broad outline, following the 1944 article 4 5 by J. L. Synge. For the specific case of the brachistochrone, we will show below that the search for Hamilton's principal function leads directly to the discovery of the differential-geometric structure underlying the problem. Synge wrote that To Hamilton, optics and dynamics were merely two aspects of the calculus of variations. He was not interested in experiments. To him, optics was the investigation of the mathematical properties of of curves giving

137

stationary values to an integral of the type

v= v x y z

j ( > ' '%%d£)du-

(6 6 1}

--

Hamilton was, in fact, a great contributor—probably the gratest single contributor of all time—to the calculus of variations. Consider two points, A' with coordinates x',y',z' and A with coordinates x,y,z. Consider all possible curves connecting A' and A. To each curve corresponds a value of the above integral. [Note by HJS&JCW: Hamilton was working with Lagrangians that are homogeneous of degree one with respect to the velocities. In that case, the choice of the interval of definition of the curves is immaterial, since every curve can be reparametrized to be defined on [0,1] without changing the value of the cost integral.] Now compare these values. Is there one curve that gives a smaller value to the integral than all others? It would seem that there must be, but it would be a rash conclusion. Suffice it to say that "in general" there is a curve giving a smallest value to the integral. Then such a curve is a ray in optics. In the language of the calculus of variations, it is an extremal. Now here is Hamilton's great central idea. Regard the minimum (or stationary) value of the integral as a function of the six coordinates x',y',z',x,y,z of A' and A. This function is Hamilton's characteristic or principal function. There is nothing hard to understand about that. What is hard to follow is Hamilton's plan to develop all the properties of the extremals from the characteristic function. In fact, it is so hard that few mathematicians in the ensuing century have given serious consideration to this plan. With the rise of optimal control, "Hamilton's plan to develop all the properties of the extremals from the characteristic function" has become one of the best and most widely used tools in the modern analysis of optimal control problems, under the name of the "dynamic programming" approach. Synge's complaint that "few mathematicians in the ensuing century have given serious consideration to this plan" may have been true in 1944 but is certainly not true today. 6.3

What is Hamilton's principal function for the brachistochrone?

Let us look at the brachistochrone problem from the modern dynamic programming point of view. The natural questions to ask are

138

1. what is the state space for the brachistochrone problem, and 2. what is the problem's "principal function" in Hamilton's sense? More precisely, we would like to know 3. what, exactly, is the controlled dynamical system q=f(q,u),

q&Q,

uGU

(6.6.2)

that corresponds to the brachistochrone problem ? (and, in particular, what is the system's state space Q ?) 4. what is the cost functional

J?

and 5. what is the value function V, regarded as a function QxQ3

fe,

qt) •->• V(qi,

? i

) € l U {-00, +00}

of the initial and terminal points, as in Hamilton's

definition?

Remark 6.1. Naturally, Hamilton's definition and Synge's version of it have to be modified to adjust them to contemporary standards of mathematical precision. Synge's text quoted above seems to require the existence of an optimum for the principal function to be well defined. We take the point of view that V(qi,qt) is always well defined as an extended real number—i.e., a member o / I U {—oo, +oo}. Precisely, V(qi, qt) is the infimum of the costs of all the trajectories going from qt to qt. In particular, V(qi,qt) = +oo if and only if there exist no trajectories from qi to qt with cost < +oo. 0 We want to address these questions taking seriously the facts that energy conservation need not be known for it to be possible to state the problem mathematically, and that, consequently, there should not be a different brachistochrone problem for each value E of the energy, but only one brachistochrone problem. It is clear that J should be time. But what exactly should we choose for Q, U, and / ? Can we, for example, take Q = E 2 ? This would mean that one can specify arbitrary points qi and qt in R2 and define V(qi,qt) unambiguously. But we know that this is not so. Given qt and qt, the minimization problem as formulated in 3.1 only makes sense for a fixed value E of the energy, and only if both qt and qt belong to the E-dependent closed half-plane H+l — I.

139

Therefore there is a well defined value VE(qi,qt) for each value of E such that qt € H+ (jj and qt € H+ (—), but these values are in general different, and the infimum of all of them is zero. A common way to get around this difficulty is by stipulating that we are only interested in dropping the particle with zero initial kinetic energy. (One can even interpret Johann Bernoulli's statement of the problem, quoted in 2.1, as suggesting this possibility.) This, however, will not do, because it would have the following unnatural consequence: if we pick two points q, q' in an optimal arc T going from qi to qt, and use T(q,q') to denote the piece of T from q to q', then T(q,q') will not in general be an optimal arc from q to q' (because the particle that started falling with zero velocity at qt will not in general have zero velocity at q). In modern terminology, the problem with zero initial kinetic energy does not satisfy the principle of optimality.d Perhaps the state space is bigger than the plane? In other words, it might be the case that one can specify the position and something else at the starting time and a similar object at the terminal time. This will not do either. We cannot, for example, specify the initial and terminal positions and velocity vectors, or even just the positions and the lengths of the velocities. What can be specified is qi, qt, and the energy—or something yielding equivalent information, e.g. the length of the initial velocity vector—and from this there is no way to derive a state space S such that V is defined on S. The key problem is that Hamilton's principal function must be a function of an even number of parameters, since it is a function of the initial and terminal states, but the natural number of variables for the value function of the brachistochrone problem seems to be five, namely, two coordinates for the initial position, two for the terminal position, and one for the energy. 6.4

Lie brackets, integrals of motion, and the differential-geometric perspective

A much better way to pose the problem is to write down the equations of motion that truly correspond to Johann Bernoulli's problem as he stated it in June 1696. One should then let the mathematical analysis lead us to the discovery of energy conservation and the simplification resulting from it. This will lead us naturally to a four-dimensional problem whose analysis is best carried out from the differential-geometric perspective, by means of Lied

J a k o b Bernoulli's solution of the brachistochrone problem makes heavy use of the fact that a portion of a minimizer is itself a minimizer. So Jakob cannot possibly have been thinking that one should only consider the case of zero initial kinetic energy.

140

algebraic calculations. The true equations of motion for the brachistochrone are X =

v, w, V = —uw w = uv y=

(6.6.3)

Here x, y, v and w are the state variables, and u is the control. The values of u are allowed to be arbitrary real numbers. The variables x and y are the horizontal and vertical coordinates of our moving point, and v, w are the components of the velocity vector. The requirement that the point is freely falling "under the influence of its own weight" means that the force effectively acting on the point should be equal to a vector proportional to [0, — g] plus a "virtual force" that does no work, i.e., is perpendicular to the velocity. Equation (6.6.3) captures these requirements, by introducing a virtual force vector of the form [uw, — uv], where u is an arbitrary "control," taking values in E. Using "t" to denote "transpose," and writing q = [x, y, v, w]^, we get a familiar equation, namely, q = F(q) + uG(q),

(6.6.4)

where F = [v,w,0, —gtf, and G = [0,0, — w,v]^. This is a 4-dimensional system, with state space Q equal to M4, and control space U equal to E. 6.5

The diamond Lie algebra

Having expressed our equations of motion in the form (6.6.4), the first thing that one should do, from the differential-geometric perspective, is to compute the iterated Lie brackets of F and G, and determine the Lie algebra structure of our problem. Precisely, we use L to denote the "accessibility Lie algebra" of our system, that is, the Lie algebra of vector fields generated by F and G. Our first task will then be to compute L. It turns out that the four vector fields F, G, [F, G] and [F, [F, G}} are linearly independent over M. (that is, there exists no nontrivial linear combination of them with constant coefficients that vanishes identically), and in addition the relations

141

[G,[F,G]]=F,

(6.6.5)

[F, [F, [F, G}}} = [G, [F, [F, G}]\ = 0

(6.6.6)

hold. It then follows easily that L is four-dimensional. As an abstract Lie algebra, L is isomorphic to the diamond Lie algebra G4, a solvable Lie algebra that plays an important role in representation theory (cf. 4 , p. 59). This Lie algebra has a basis H,P,Q, E, with the commutation relations [H,P}= [H,Q] =

Q, -P,

[P,Q]= [HtE\=

E, [P,E} = [Q,E} = 0.

(To construct an isomorphism $ from G4 to L, define * ( P ) = [F,G], *(Q) = F, *(ff) = G,*(E)

= ~[F, F,G\] •

A concrete realization of G4 as an algebra of differential operators—acting on smooth functions on the real line—can be defined by "quantizing the harmonic oscillator," i.e., by taking P Q E H

dx ' multiplication by x, identity,

i<-PW) = i(-^ + * ! ),

so that H is the Hermite operator. 6.6

Energy conservation

We now compute the accessibility distribution A generated by L. Formally, q i-> A(q) is the map that assigns to each state q the linear space A(q) = {X(q)

:XEL}.

To find A we first observe that, even though the vector fields F, G, [F, G] and [F, [F, G]} are linearly independent over R, a simple computation shows that G = <;1F + <;2[F,G} +

<;3[F,[F,G}},

(6.6.7)

142

where the Q are smooth functions of q, given by v

Ci=—,

9

w

C2 =

9

, Cs =

v2 +w2 5-2-. V

6.6.8

From these facts one can easily show that every iterated Lie bracket of F and G is a linear combination of F, [F,G], and [F,[F,G]] with smooth coefficients. Since F(q), [F,G](q), and [F, [F,G]](q) are linearly independent at each point q E E 4 , we can conclude that A is 3-dimensional at each point. This means—using, for example, Probenius' theorem on the existence of integral manifolds—that, at least locally, there is a nontrivial integral of motion, i.e., a function with nonzero gradient which is constant along all integral curves of F and G, and then also along all solutions of (6.6.4). This integral of motion can be easily computed and turns out to be the energy E, given by E{x,y,v,w)

v2 + w2 = — - — + gy.

(6.6.9)

Notice that E is a smooth function on E 4 , with nowhere vanishing gradient. Therefore the level hypersurfaces of E—i.e., the sets SE^

{q : E(q) = E] ,

for E € E—are smooth 3-dimensional manifolds. (The set SE is given by the equation V=

2E - v2 - w2 Tg '

which shows that each SE is the graph of a smooth function (x,v,w) 6.7

Naive application of the Maximum

H-> y.)

Principle

The fact that our 4-dimensional state space E 4 is foliated by 3-dimensional "leaves" SE has profound implications for the optimization problem. Indeed, if we try to find the optimal trajectories by applying the Maximum Principle, we get nothing. This is a consequence of more general result, namely, that, ( # ) whenever a control system satisfies a nontrivial "holonomic constraint" such as E = constant, then the Maximum Principle is uninformative, because every trajectory is an extremal. (The word "nontrivial" here means that VE never vanishes.)

143

Remark 6.2. The reason for ( # ) is as follows. The necessary condition given by the Maximum Principle for a trajectory-control pair (£, n) to be optimal is that (£, n) be either a "normal extremal" or an "abnormal extremal." Moreover, being an abnormal extremal is a necessary condition for the pair (£, n) to be such that the terminal point of £ does not belong to the interior of the reachable set from the initial point. If a nontrivial constraint such as E = constant is satisfied, then the reachable set from any initial point q is contained in a submanifold of the state space of positive codimension. Hence the reachable set has empty interior. Therefore all trajectory-control pairs are abnormal extremals. ^ 6.8

Controllability

We now investigate the controllability properties of our system. First of all, it is clear that two points qi, qt cannot be joined by a trajectory of (6.6.3) unless they belong to the same leaf SE- We now show that the converse is true, that is, any two points qi, qt belonging to the same leaf of (6.6.3) can be joined by a trajectory. In other words, the restriction of (6.6.3) to each leaf is completely controllable. To prove our statement, we use the technique of "Lie saturates" introduced by Jurdjevic and Kupka. Fix E and let (6.6.3.E) denote the restriction of (6.6.3) to SE- By construction, the system (6.6.3..E) has the accessibility property. So to prove complete controllability it suffices to establish complete controllability of the Lie saturate family of vector fields C. It is clear that F, G and — G belong to C. So the desired conclusion will follow if we show that — F G C. But this follows easily, because the identity [G, [G, F]\ = — F implies e*GFe-*G

6.9

=

_

p

The existence problem

A precise definition of Hamilton's principal function V was given in Remark 6.1. It is clear that, if qi and qt belong to R 4 , then V(qt, qt) < +00 if and only if qi and qt belong to the same leaf SEIt is then reasonable to ask whether an optimal control steering qi to qt exists whenever V(qt,qt) < +00. Equivalently, we want to know if the infimum that occurs in the definition of V is in fact a minimum whenever it is finite. The usual way to prove existence of an optimum is to pick a sequence {??j}j£N of controls steering qi to qt with costs Cj that converge to the infi-

144

mum, look at the corresponding trajectories £,, and then try to produce a subsequence {r)j(e)}e€T$ of {^}jgN that converges in some sense to a limiting control TJOO and is such that the £,•(£) converge to a trajectory £00 of rjco and the costs Cj(^) converge to the cost CQQ of (£oo,»?oo)This method may fail for various reasons. The simplest one is that the £j may "go off to infinity." That is, there may not exist a subsequence {£j(e)}teN that stays in a fixed compact set. In the following definition, we give a name to the property that this particular obstruction to the existence of optimal controls does not occur. Definition 6.3. We say that an optimal control problem with dynamics (6.6.2) and Lagrangian cost functional with Lagrangian L satisfies the properness condition if for every pair qi,qt of states such that V(qi,qt) < +00 there exist a number c € l and a compact subset K of Q such that (1) V(quqt)


(2) whenever rj is a control steering qi to qt with cost < c, and £ is the corresponding trajectory, it follows that £ is entirely contained on K. 0 When a problem satisfies the properness condition, it still does not follow that optimal controls exist, because the minimizing sequences {T]J}J€N need not have subsequences that converge in any reasonable sense. On the other hand, it is commonly agreed that in this case optimal controls really exist if one enlarges the original class U of admissible controls by adding to it some "generalized controls." We will show that this is indeed what happens for our brachistochrone problem, and that in this case the generalized controls can be easily described in concrete terms, and turn out to be none other than "impulse controls." To begin with, we prove the following. Proposition 6.4. The minimum time problem for the brachistochrone system (6.6.3) satisfies the properness condition. Proof. Let qi, qt be two points of M4 having the same energy E. Fix an arbitrary c € E such that c > V(qi,qt). We have to find a constant A such that ||£(i)|| < A for all t € [0,r] whenever r < c and £ : [0,r] (->• K4 is a trajectory of (6.6.3) for some control t 1-4 r)(t) such that £(0) = qi and &T) = qt. Write £(*) = [x{t),y(t), v(t),w(t)]*. Let

9(t) = [v(t)Mt)V, at) = Kt)M*)V-

145

Then (6.6.3) implies that dt

|| C (t)|| 2 = - 2

9

•«,(*).

Therefore dt

2 2 IIC(*)IIS = 2g\w(t)\<9 +w(t)
+\\at)\\

and then, if we write

V>(*) = |llC(*)ll 2 -IK(o)|| 2 |, we find that < [\g2 + \K(s)\\2) ds < c(g2 + ||C(0)||2) + / V(*) ds. Jo Jo Then Gronwall's inequality implies that m

W)

< c(g2 + | | C ( 0 ) | | V < c{g2 + ||C(0)|| 2 )e c .

It then follows that

IICWII 2
0 6.10

• Application of the Maximum Principle on manifolds

Using the integral of motion E, we can regard E 4 as foliated by the 3dimensional level hypersurfaces of E, and conclude that every trajectory is contained in one of the leaves Sg. If E G E, then the Maximum Principle on manifolds (cf. 3T ) can be applied to the problem restricted to Sg. The conclusion turns out to be exactly the same as that for the unrestricted problem, except that the nontriviality condition for the momentum is slightly stronger. This extra strength is actually fundamental for, as explained in 6.7, without it the conclusion would be completely uninformative. If [a, b] 3 t H-> £„(£) € E 4 is an optimal trajectory for our problem, corresponding to a control [a, b] 3 t i-> n* (t) € E, then the usual version of the Maximum Principle yields the existence of a nontrivial momentum vector field [a, b] 3 t !->• 7r(t) 6 E 4

146

(often called an "adjoint vector") that satisfies the adjoint equation, the Hamiltonian maximization condition, and the additional condition that the maximized value of the Hamiltonian is a nonnegative constant 6 . The principle applied to the restriction to the leaves yields the stronger conclusion that n(t) cannot be orthogonal to the leaf Sg that contains £*. (In differential-geometric terms, -?v(t) is really a covectoron Sg at £*(£), and has to be nontrivial as such. Equivalently, if one insists on regarding 7r(i) as a vector in R 4 , then it should not be orthogonal to the tangent space to Sg at £*(£).) (Naturally, one could also avoid using the Maximum Principle on manifolds by invoking instead the fact that Sg is identified with M3 in an obvious way, since it is the graph of a smooth function (x,v,w) i-> y(x,v,w). This would amount to eliminating y from the equations by substituting for it its expression in terms of the other variables, for each fixed E.) The Hamiltonian is the function R4 x E x M4 3 (q,u,p) >-> R defined by H(q,u,p)

=
(6.6.10)

where the functions ip, ip are defined by 1>{q,p) =

•


Since u is completely unrestricted, H cannot have a maximum as a function of u unless tp(q,p) = 0. So V"(6(*).T(*)) = 0

for

a11

*•

(6.6.11)

t,

(6.6.12)

Differentiation of (6.6.12) yields p(^(t),Tr{t))

= 0 for all

where p{q,p)d=
147

A further differentiation—together with the fact that [G, [F, G]] = F—yields 0(M*)Mt))

+u(t)
= 0,

(6.6.13)

where e(q,p)^{p,[F,[F,G]](q)). This, together with the equalities ip = p = 0, determine u as a function of q, i.e., provide an optimal control in the form of a state space feedback law u = u(q), as we now show. First of all, (6.6.7) implies that *l> =
(6.6.14)

along our trajectory. Next, we use the crucial fact that ¥>(*(«),&(t)) 5* 0,

(6.6.15)

which follows from the stronger version of the Maximum Principle. (The precise argument is as follows. The covector n(t) must be nontrivial as a linear functional on the tangent space at £*(<) of the leaf containing £*(t). Since this space is spanned by the three vectors F(^(t)), [F,G](£*(£)) and [F,[F,G]](Ut)), one of the numbers ¥>(*(*),&(*)). p{*(t),t.{t)), 0(TT (*),£,(*)) must be nonzero. We know that p(ir(t),t;*(t)) = 0. If <^(7r(t), £*(*)) = 0, then (6.6.13) would imply that 6{n(t), £» (£)) = 0, and we would get a contradiction. Hence (6.6.15) holds.) Therefore (6.6.13) implies u{t)

(6 6 16)

—
--

that is U{t)

~

(«(t),F(Ut)))

(6 6 17)

•

- -

Moreover, it follows from (6.6.14), together with the fact that I/J = p = 0 along our trajectory, that Ci(Ut)M^(t),^(t)) Then (6.6.16) implies

+ C3(Ut))0(At),Ut))

= 0.

(6.6.18)

148

as long as C3 (£*(*)) ^ 0. Therefore u{t) = u(Ut)),

(6.6.20)

where , def

Ivg

uj(q) = u{x, y, v, w) =

(6.6.21) 2 2 . v2 + w'z It is then clear that u is a smooth function on the complement of the set S = {(x,y,v,w)

:v2+w2

= 0} .

(6.6.22)

The "singular" set £ corresponds, naturally, to the points where the kinetic energy vanishes, which as we know from our other analyses is where some singular behavior occurs. If we define Z(q)=F(q)+Lj(q)G(q), we see that Z is a smooth vector field extremals in the sense of the Maximum The resulting trajectories are easily use the feedback u = u(q), the dynamic

(6.6.23) 4

on M \E whose trajectories are the Principle. computed. First, observe that, if we equations become

x = v, Vw, v = —2g(v2 + w2)~1vw , 2 1 2 L ...2w..2 w = 2g(v22_+w )~ v —„ g 2 2 1 2 = {v + w )' {2gv - g{v2 + w2)) = g{v2 +w2)~1(v2 -w2).

(6-6-24)

If we introduce a complex variable z denned by z = v + iw, then the last two equations of (6.6.24) say that

i = ^ 1

that is, z=9—.

(6.6.25)

z

\\ '* The formulas of (6.6.25) define a well known dynamical system of the complex plane C. The right-hand side of the equations is a smooth vector field on C\{0}. The integral curves are circles passing through the origin and having center on the real axis (plus the imaginary axis, parametrized by E 3 t —> (0, — gt), which is also a solution since along it (6.6.25) just says that i = — gi). The parametric equation of the circles is z(t) = r{l + a e^), where a £ { + 1 , - 1 } and r is a positive real number.

(6.6.26)

149

Once it has been shown that the velocity vector {v,w) evolves according to (6.6.26), a trivial integration shows that the trajectories described by the position vector (x, y) are cycloids. 6.11

The complete synthesis and Hamilton's principal function for the 4-dimensional brachistochrone problem

We haven't yet completely answered all our earlier questions and determined Hamilton's principal function V. Let qi, qt be two points of R4. It is clear that V(qi, qt) = +oo unless both <7i and qt belong to the same energy hypersurface, because if the energies are different then the points cannot be joined by a trajectory, and the infimum of the empty set is +oo. On the other hand, if the points qi and qt belong to the same level hypersurface Sg, then we know that they can be joined by a trajectory, so V(qi, qt) will be finite. Moreover, it is reasonable to expect that such a curve will exist, and if it does not then we ought to be able to explain why. Write qi = (Ai,a,i), qt = (At,at), where Ai} at, At, at, belong to R 2 . The classical analysis carried out earlier seems to suggest that, once Ai, At and E are specified, there will be only one optimal arc joining them. So, if we specify Ai and At but do not fix the energy, then there should be a 1parameter family of such arcs (one arc for each value of E). Yet we now seem to need a 3-parameter family, because we want to accommodate all possible pairs (oj, at) (i.e., four parameters) as long as they give rise to the same energy (one constraint, and 4 — 1 = 3 ) . Equivalently, given E, we ought to be able to choose the directions 0i, Ot, of the velocity vectors ai and at in a completely arbitrary way. (Their magnitudes are determined by the fact that the energy at qi and qt must equal E.) But given Ai, At and E there is a unique cycloid from Ai to At corresponding to the energy E, and in particular the directions 6i, 6t at Ai, At of this cycloid are completely determined by Ai, At and E. In other words, we ought to be able to choose freely the two angles 9i, 6t, but it appears that we cannot do so. The solution of this apparent paradox is as follows. Suppose we fix an energy level E. Given any Ai, the set C(Ai, E) of those a's such that (Ai, a) € Sg is either empty or a circle, except for the "borderline" case when (Ai,0) £ Sg and C(Ai,E) consists of a single point. If C(Ai,E) is a circle, then through each point of {Ai} x C(Ai,E) there passes an integral curve of Z. The set of all points lying on such curves is 2-dimensional, and intersects the 1-dimensional set {At} x C(At,E) at just one point. This means that there is a unique pair (a.i,at) such that (Ai,a,i) and (At,at) lie on the same integral

150

curve of Z. The portion of that curve joining (At, en) to (At,at) is, of course, our famous cycloid. What is then the optimal trajectory joining (Ai, at) to (At, at) for general ai, at"? What is the corresponding value of V(Ai,ai,At,at)'? The answer is quite simple: the "optimal trajectory" is obtained by 1. applying an impulse control to rotate a; to a; instantaneously, 2. then following the cycloid from (Ai,cn) to

(At,at),

3. finally applying another impulse control to rotate at to at. (These are not true trajectories, of course, but it is easy to translate the above into a statement about a minimizing sequence of true trajectories.) The value V(Ai,cii, At,at) is therefore equal to V(Ai,a,i,At,dt). 7 7.1

Five modern variations on the theme of the brachistochrone Hestenes' "variation of the brachistochrone problem"

In 1956, M. Hestenes wrote (in

18

, p. 64):

... it is interesting to speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times. An interesting variation of the brachistochrone problem can be described as follows: Consider an idealized rocket ship moving in a vertical plane. The ship is to be considered as a particle acted upon by a gravitational force g and a thrust of constant magnitude F and variable inclination (p ... There are no other forces acting on this ship. The pilot steers the ship by controlling the inclination
y = y(t)

0
subject to the constraints (with unit mass) x — F cos tp

y = F sin (p — g

and with x, y, x, y, prescribed at t = 0 and t — T, one which minimizes the time of flight T.

151

It can be shown that the path of least time has the following interesting property: there is a line L with direction fixed in space, and with position fixed relative to the ship, such that if a particle M is moving along L with constant speed, the ship will traverse a path of least time by keeping the thrust aimed at the particle M. Hestenes later on proceeds to solve the problem. Let us present a solution based on optimal control. The dynamical equations can be rewritten as x =z

z = F cos ip

y =w

w = F sin (p — g.

If [0,T] 9 ( 4 (x*(t), z*(t),yt(t),w*(t)) e I 2 is an optimal trajectory, corresponding to a control function [0, T] 3 t t-> ip* (t) € M, then the Maximum Principle yields a nonnegative constant -KQ and and a vector-valued function [0,T]3t^n(t)

=

(nx(t),irz(t),Try(t),Trw(t)).

The Hamiltonian is given by H(x,z,y,w,ip,px,pz,py,pw,po)

= px z+pz Fcosip+py w+pw (Fsmtp-g)-p0

.

The adjoint equation becomes •kx = 0

7rz = — 7rx

Then, if we write u)(t) — {nz(t),Ttw(t)), u(t) =

-ky = 0

nw = — •Ky .

we have tui,

(JQ +

where u>o and u>i are constant two-dimensional vectors. The Hamiltonian maximization condition then says that the vector u(i) = (cos ip(t), sin ip(t)) is given by

""» = »

•

<"•»

The Hamiltonian must vanish along our trajectory. Therefore, ir(t) can never vanish, for if 7r(t) = 0 it would follow that TT0 = 0, contradicting the nontriviality condition. Hence u>(t) cannot vanish identically. So u is a linear function which is not identically equal to zero. Therefore there is at most one value of t such that uj(t) = 0. It follows that the function u of (7.7.1) is well denned except possibly at one point i. Moreover, if such a i exists, then cj(t) = (t — i)cji, so if t < t u(t) = {—wi ui\ if t > i.

152

The "line L" of Hestenes' "interesting property" is the line Lt with direction <jj\, with a point At having a "position fixed relative to the ship" given by At = (x*{t),y*(t))

+u}0.

So in fact Lt is moving along with the ship. If a point qt on Lt moves with constant velocity UJ\, starting at Ao at time 0, then the point's position at time t is A* + toj\. Therefore the vector from the ship's position to qt is precisely u)(t). We have shown that the direction of the thrust is uj(t), from which Hestenes' "geometric property" follows. 1.2

The reflected brachistochrone problem

In 3, our analysis of the brachistochrone problem was carried out in a halfplane, by looking only for optimal trajectories that are entirely above the x axis, as in Johann Bernoulli's original problem. Let us pursue Hestenes' idea that "it is interesting to speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times." An obvious question for the brothers to ask would have been that of finding out what happens if we work in the whole plane. This is in fact a much more natural mathematical setting for the brachistochrone problem. We shall refer to this new question as the reflected brachistochrone problem. (It is because we were thinking all along of this extension that we wrote y/\y~\ rather that ^Jy in (5.5.1)). For a recent study of the reflected brachistochrone problem from the dynamic programming perspective, see Malisoff 2 8 ' 2 9 . Solving this more general problem amounts to finding the light rays when the medium is the whole plane, and the speed of light is ^/\y\. In other words, we want to find the minimum time paths for the control system (5.5.1) with control set U given by (5.5.2) and state space equal to the whole plane. Notice that this problem is "completely controllable," in the sense that (C) any two points A, B of R2 can be joined by a feasible path, even if they lie on opposite sides of the x axis. To prove (C), observe first that the desired conclusion is trivial if A and B both lie in the open upper half plane or in the open lower half plane. So it clearly suffices to prove the conclusion if A lies in the open upper half plane and B belongs to the x axis. Moreover, we can assume that A lies on the same vertical line as B, for otherwise we can pick A' on the same vertical as B but strictly above B, and join A to A' first in finite time. So let us suppose

153

that A - (a,a) and B{a,0). Then the curve [0,a] 9 s 4 ( a , a - s ) certainly goes from A to B. Along this curve, dy = -^/ydt, so dy

ds

Therefore the total time T required to move from A to B along this curve is given by fa J0

ds

n

r-

yfa-s

so T < +oo, and (C) is proved. The crucial point here is that the right-hand side of (5.5.1) vanishes along the x axis, but this does not prevent the existence of feasible paths crossing the x-axis in finite time, because the "speed of light function" (x,y) <-+c(x,y) = y/\y] is not Lipschitz near the x axis. Remark 7.1. If the function c was Lipschitz-continuous, then the well known uniqueness theorem for ordinary differential equations would imply that every solution passing through a point on the x axis is a constant curve. This would obviously render it impossible for feasible curves starting at a point not on the x axis to reach the x axis in finite time. An important example of this situation is the case when the speed o light is \y\ rather than y/\y\. That case was also studied by Johann Bernoulli, as is clearly seen in the text we quoted at the end of 2.2, where he says that if, for example, the velocities were as the altitudes, then both curves [HJS & JCW: that is, the brachistochrone and the tautochrone] would be algebraic, the one a circle, the other one a straight line. It turns out that the "brachistochrones" for the case when the speed of light is \y\ are half circles whose center lies on the x axis. Moreover, these circles take infinite time to reach the x axis, precisely because the function J/ — ' >• |y| is Lipschitz. ^ We have seen that the complete controllability of (5.5.2) depends essentially on the non-Lipschitzian character of its speed of light function. However, this same non-Lipschitzian character also renders the Maximum Principle inapplicable, in its classical and nonsmooth versions (including the so-called "Lojasiewicz version," cf. Sussmann 35), since all these results require a Lipschitz reference vector field. Can we still use necessary conditions to find the solution, perhaps by invoking a more general version of the Maximum Principle that would be applicable to this nonsmooth, non-Lipschitzian case?

154

Suppose, for example, that we want to find an optimal trajectory £ going from A to B, where A lies in the upper half-plane and B is in the lower halfplane. Then one can easily show, first of all, that an optimal trajectory £ exists, using the Ascoli-Arzela theorem as before (cf. 5.2). Next, using the usual necessary conditions for optimality, e.g. the Euler-Lagrange equation or the classical version of the Maximum Principle, one shows that any portion of an optimal curve which is entirely contained in the closed upper half plane or in the closed lower half plane is a cycloid given by (3.3.5), or a reflection or such a cycloid with respect to the x axis. Next, one sees that £ cannot traverse the x axis more than once. (This is a very straightforward consequence of Lemma 5.1.) Therefore £ must consist of a cycloid T\ going from A to a point X in the x axis, followed by a second cycloid 1^ going from X to B. It remains to find X. Naturally, X is the only point on £ in a neighborhood of which the usual smooth and nonsmooth versions of the Maximum Principle are inapplicable. On the other hand, it is easy to produce a nonrigorous formal argument yielding the missing condition, as we now show. Let us apply the Maximum Principle formally, even though the technical conditions required for the validity of the usual versions are not verified. The maximized value of the Hamiltonian is clearly equal to ||p||\/[y| — po, and we know that this number is in fact equal to zero. So \\p\t)\\V\y\=Po

for all

t.

(7.7.2)

If we multiply (5.5.6) by \y(t)\, and use (7.7.2), we find

|y(*)l ( l + ^ ) » ) = M * ^ K = ^ .

(7.7.3)

The adjoint system (5.5.4) tells us that p is constant on each time interval that does not contain the point r such that £(r) = X. If we make the Ansatz that p is actually constant on the whole time interval, i.e., that the constant values of p on both sides of T are equal, we find that the cycloids Ti, T2 must be such that the quantity C=\y(x)\(l+y'(x)2).

(7.7.4)

which is constant along each Tj, actually has the same value for both curves. This extra condition determines X. The above argument is, of course, not rigorous. It turns out, however, that the recent "very nonsmooth" versions of the Maximum Principle developed by one of us—cf. Sussmann 39>4o,4i,42,43,44—apply to this problem and provide a

155

rigorous justification for our formal argument. The reason for this is that the new results do not require Lipschitz continuity—or even continuity—of the reference vector field. All that is needed is that the flow maps generated by the reference flow be differentiable in an appropriate sensed Remarkably, our "extra condition" has a simple geometric meaning. Recall that the cycloids of Johann Bernoulli's problem are generated by a circle rolling along the x axis. The extra condition then says that

(C) The rolling circles that generate the upper and lower cycloids must have equall radii. 7.3 A modest claim If we are going to "speculate" as Hestenes did, it is natural to ask not only what questions the Bernoulli brothers would have asked if they had lived in modern times, but also what kinds of answers they would have liked. This is of course a hard question to answer, and anything we say will have to be highly subjective. Granted this, we are firmly convinced that 1. The Bernoulli brothers would have liked the "interesting property" of the solution of Hestenes' problem (which is almost certainly the reason why Hestenes chose to emphasize this property) and also that 2. they would have found elegant the geometric condition of our solution of the reflected brachistochrone problem. This seems clear to us but, since de gustibus non est disputandum, we shall not pursue this particular argument further. 7.4

The brachistochrone with friction

In the problem studied in 6, we assumed that there was no friction. This implied that energy was conserved and that the four-dimensional state space was foliated by three-dimensional orbits of the system's accessibility distribution. ' F o r the specific case of the reflected brachistochrone, classical differentiability suffices. For more general problems, one needs other theories of "generalized differentiation," as e x p l a i n e d in 39,40,41,42,43,44

156 It is then natural to consider the more general case of the brachistochrone with friction, and to expect that for this problem there will be no nontrivial conserved functions of the state, so the accessibility distribution will be fourdimensional, and the system will have the accessibility property. We now outline some of the results, without carrying out a complete analysis. The dynamical behavior of the system is given by the equations x—

v, w, v = —uw — pv, w = uv — pw — g, y =

(7.7.5)

where, as before, u is an unbounded scalar control. The parameter p is a real number, the friction coefficient. The problem with friction corresponds to p > 0. When p = 0 we get the frictionless case considered earlier. If we write

F(q) =

v w -pv -pw

G(q) =

0 0 -w v

then (7.7.5) takes the form q=

F(q)+uG(q).

(7.7.6)

From now on, we assume that p ^ 0, since the case when p — 0 has already been discussed. Let L be the Lie algebra of vector fields generated by F and G. In order to analyze the structure of L, it will be convenient to introduce five new vector fields Z, V, W, X, and Y. The following eight formulas express F, G, and [G, F] in terms of the usual differential operator notation for vector fields, and define Z, V, W, X, and V:

F = vdx + wdv — pvdv - {pw + g)du vd„ G= - wdv + [G,F] = -wdx + vdy — gdv wdu Z= vdv + V = dv

w = X = Y =

dx 8y

157

Let S be the linear span of the eight vector fields F, G, [G,F], Z, V, W, X, and Y. It then turns out that S C I , because Z = a(F + [G,[G,F}}), W = 7 ([F, Z] + F + PZ), V = [G,W], X = pV-[F,V], Y = PW- [F, W], where 1 , 1 a = - and 7 = — . P 29 On the other hand, the formulas [F, [G, F]] = -p[G, F] + 2gX - 2pgV, [G,[G,F]] = -F-pZ, [F,Z] = [F,V} [F,W] [F,X] [G,Z] [G,V] [G,W]

= = = = = =

-F-pZ-2gW, -X + pV, -Y + pW, [F,Y] = 0, [G,X] = [G,Y] = 0, -W, V,

show that [F, S] C S and [G, S]CS. So L = S. A straightforward calculation shows that F, G, [G,F], Z, V, W, X, and Y are linearly independent over E. So L is an S-dimensional Lie algebra. Let A be the accessibility distribution of our system, so A(q) = {h(q) : h 6 L} if q 6 K4. Since the vector fields X, Y, V and W belong to L, it is clear that A(q) = M4 for every q. So our system has the accessibility property. An analysis similar to the one carried out in the previous section for the frictionless problem yields an optimal "feedback control" 2gv _Cy2 ~ vw u = — - — 5 - + 2 np — 25 - — r2> v + w involving an additional parameter £, given by _ TVy_

TTx

{7.7.7)

158

(Here irx, ny are the X and Y components of the momentum, which are easily seen to be constant along extremals.) In other words, through every point q € E 4 there passes a one-parameter family of optimal trajectories, depending on the parameter C,. 7.5

The classical brachistochrone as a degenerate problem

The family of brachistochrone problems with friction coefficient p constitutes an "unfolding" of the classical frictionless problem. For p ^ 0, the situation is generic, in the sense that the accessibility distribution is four-dimensional. The generic behavior of the time-optimal trajectories for such problems is precisely the one described here: the control is given in "feedback" form, depending on the state and an extra parameter. Formula (7.7.7) shows that the extra parameter only occurs as part of a term that is multiplied by p. It follows that when p = 0 the extra parameter disappears, and we get a state space feedback as determined earlier. 7.6

Deriving Snell's law from the Maximum Principle

Our third "modern variation" will involve going back to ancient times, long before 1696, and considering Fermat's minimum time principle and Snell's law of refraction. We have already explained how Snell's law is intimately related to the brachistochrone problem, since it played a crucial role in Johann Bernoulli's solution. So, if we pursue Hestenes' line of inquiry further, and speculate on other questions that the Bernoulli brothers might have asked "had they lived in modern times," it is easy to imagine them asking the following. (Q) Given that in these modern times powerful general theories have been developed, going far beyond the particular examples that were dealt with by ad hoc methods up to the 17th and early 18th centuries, do these general theories actually cover all these special examples whose study preceded and begat them? As an important special case of this question, the brothers would certainly want to know whether the necessary conditions of the calculus of variations and optimal control theory apply, in particular, to the derivation of Snell's law. So let us consider a medium such as the one we described when discussing Snell's law, in which the speed of light is constant on each of the two sides H-, H+ of a hyperplane H in K n , but the constant values c_, c+, are different.

159

Let us ask the obvious question, namely, whether Snell's law be derived from the Maximum Principle. It is easy to see that there is no way to formulate this question as a minimum time problem with a controlled dynamics q = f(q,u,t) in which / is continuous with respect to q, since | | / | | has to jump as we cross H. In the usual versions of the Maximum Principle, it is not required that / be differentiate, or even continuous, with respect to the control u. But here the situation is different, since we need to be able to deal with an / which is discontinuous with respect to the state variable q. Once again, it turns out that the recent versions the Maximum Principle developed in Sussmann 39>4o,4i,42,43,44 apply to this problem as well, and lead to Snell's law. The reason is that in these version all that is needed is that the reference trajectory £* arise from a flow {$t',t} consisting of differentiable maps, and there is no need for this flow to be generated by a continuous vector field. 7.7

Was Johann Bernoulli close to discovering non-Euclidean geometry?

Given a "speed of light function" (x,y) •-* c{x,y) on some plane region, the time along a curve is given by _ y/dx2 + dy2 c(x,y) since y ^f + -^ = c(x,y). This means that, if we adopt a slightly different point of view, and think of time as "length," then the function c defines a Riemannian metric on the set fi c = {{x,y) : c(x,y) > 0}, provided that c is sufficiently smooth. Let us assume that c(x,y) > 0 whenever y > 0, as was the case in our two examples. Then c gives rise to a Riemannian metric 2

_ dx2 + dy2 c(x,y)2

on the upper half plane. The "brachistochrones" are then the minimum length curves, that is, the geodesies. In the case of the "true brachistochrone," when c(x,y) = y/\y~\, these geodesies are cycloids (plus vertical segments) but maximal geodesies have finite length. On the other hand, in the case when c{x, y) = \y\, so that the Riemannian metric on the upper half plane is given by ds2=y-2(dx2+dy2), the geodesies are, as we have seen, half circles of infinite length.

(7.7.8)

So, using modern terminology, the circles that occur as solutions of this modified version of Johann Bernoulli's brachistochrone problem are the geodesies of a Riemannian metric on the upper half plane given by (7.7.8). We now know that this Riemannian metric defines the so called Poincare half plane, which is the simplest model of non-Euclidean geometry. This raises an obvious question: how close did Johann Bernoulli get to solving the famous problem of the provability of Euclid's fifth postulate? What renders the question particularly interesting is that 1. Johann Bernoulli lived at a time when the problem of Euclid's parallel postulate was very much at the center of mathematical research. For example, G. Saccheri published in 1733 his book 3 2 , entitled Euclides ab omni naevo vindicatus (that is, "Euclid vindicated from every flaw"), where he pushed very far the development of non-Euclidean geometry, only to end up dismissing it by claiming that a particular conclusion he had reached had to be rejected because it was "repugnant to the nature of a straight line." 2. The cycloids that occur in the original brachistochrone problem have many of the incidence properties of Euclidean lines, and therefore come close to satisfying all Euclid's postulates other than the fifth. But they do not yet provide a good model for non-Euclidean geometry because they "go to infinity" (i.e., approach the x axis) with finite length. 3. The half-circles of the Poincare model have all the desired incidence properties and also have infinite length, so they furnish a perfect model where all the Euclidean axioms other than the fifth are true. 4. Johann Bernoulli did study these half circles, as we pointed out in Remark 7.1. So in a sense he did discover the Poincare model. 5. Johann Bernoulli was definitely interested in the question, since he made significant contributions to the early stages of the development of the idea of geodesies on curved surfaces, by studying the curves of minimum length on a surface, which later 9 came to be known as "geodesies." 6. The solution of the problem of the independence of Euclid's postulate is extremely simple by today's standards, even though mathematicians took about 2200 years to find it. Actually, the solution is so simple that it can be explained to any undergraduate with a moderate understanding of plane analytic geometry. All that is required is to take any 9

T h e word "geodesic" was coined much later by Liouville (1809-1892).

161

reasonable way to compute "time" along curves, rename "time" by calling it "length," and then call the length-minimizing curves (that is, the "brachistochrones") "lines." For example, the case when c = \y\ leads us to taking the half circles with center on the x axis (plus the vertical half lines) and calling them "lines." Once this trivial renaming has been carried out, it is easy to verify that all of Euclid's postulates other than the fifth hold, but the fifth postulate fails. So Johann Bernoulli had essentially all the tools he needed to solve the problem and prove that Saccheri was wrong, by showing that Euclid's fifth postulate does not follows from the other ones. Why didn't he do it? One can surmise that he lacked the understanding of the geometric significance of the cycloids and half circles that he had found as solutions of his minimization problems. He certainly saw these curves as "geometric," but failed to see that they could serve as the "lines" of a different geometry. In modern terms, this is a trivial remark, namely, that one can turn around the well known fact that the segment is the shortest path between two points, and make it the definition of the notion of segment, while at the same time using a more general prescription to measure "length." But the idea that minimization problems can be the source of new geometries evolved very slowly over two millennia, and took a long time to crystallize, reaching final form in the nineteenth century, in the era of Gauss and Riemann, when it revolutionized mathematics and then moved on, at the beginning of the twentieth century, to bring a a fundamental change in our understanding of the physical universe. One can already find the idea in a rudimentary form in Fermat's least time principle'1, which can be interpreted as saying that light rays follow the shortest paths, with the proviso that "length" must now be defined in terms of a new and more refined geometry, in which the physical properties of space influence the geometry, and shortest paths can be curved or bent. This interpretation is suggested, for example, by G. Lochak, in 27 , p. 57-58 [our translation]: [Fermat's principle] anticipated the greatest revolution to be undergone by geometry in the future, namely, the discovery of non-Euclidean geometries, because it showed for the first time that the physical properties of a medium might cause the shortest paths not to be straight lines. ''which, as we have seen, was a direct ancestor of Johann Bernoulli's solution of the brachistochrone problem

162

In fact, Leibniz definitely uses the word "shortest" in his analysis of the refraction problem in his 1684 Nova methodus paper, showing that he was at least entertaining the possibility that light rays in a medium with variable speed of light truly were "shortest paths" in some new geometry. But it would appear that, in spite of these earlier glimpses of what would become one of the greatest scientific ideas of all times, the era of Johann Bernoulli was not yet ready to take the trivial step of turning around the lenght minimization property of segments. Johann Bernoulli passionately wanted to succeed and be famous, and was obsessed by the desire to surpass his contemporaries, especially his brother Jacob and his son Daniel*. It is thus sadly ironic that he, of all people, got so close to solving an important 2000-year old problem but missed because he failed to take a small step, a step so simple that today it can easily be explained to beginners. Had Johann Bernoulli "lived in modern times," he probably would have been torn between the curiosity to find out about non-Euclidean geometry and the desire not to hear about the tragedy of his missed opportunity to reach eternal fame and glory. Acknowledgements The writing of this paper began while the first author was Johann Bernoulli Visiting Professor at the University of Groningen, in May, June and July 1999, and continued while he was back at Rutgers University, partially supported by the National Science Foundation under NSF Grant DMS 98-03411. It was concluded while he was Weston Visiting Professor at the Weizmann Institute of Science in Rehovot, Israel, from October 2000 to January 2001. He is grateful to the University of Groningen, the Weizmann Institute, and NSF, for their help and support. References 1. Bell, E. T., Men of Mathematics. (Simon and Schuster, New York, 1937). 2. Beltrami, E., Saggio di interpretazione della geometria noneuclidea.Giornale di Matematiche VI, 284-312 (1868) (in Italian). "Johann was very jealous of Daniel's success. He once threw Daniel out of the house for having won a French Academy of Sciences prize for which Johannn had also been a candidate (cf. Bell l , p. 134). Probably with the help of his loyal disciple Euler (a student of Johann in Basel and a colleague of Daniel in Saint Petersburg), Johann plagiarized Daniel's pathbreaking work on hydrodynamics (cf. Guillen 1 7 ) .

163 (Eng. transl. in Stillwell, 33 , 7-34). 3. Berkovitz, L. D., Optimal Control Theory. (Springer-Verlag, New York, 1974). 4. Berna, P., N. Conze, M. Duflo, M. Levy-Nahas, M. Rais, P. Renouard, M. Vergne, Representations des groupes resolubles. (Dunod, Paris, 1972). 5. Bernoulli, Johann, Curvatura radii in diaphanis nonuniformibus . . . [The curvature of a ray in nonuniform media . . . ]. Acta Eruditorum, May, 206-211 (1697). {Opera omnia I, 187-193.) 6. Bernoulli, Jakob, Solutio problematum fraternorum . . . una cum propositione reciproca aliorum [Solution of problems of my brother . . . together with the proposition of others in turn]. Acta Eruditorum, May, 211-217 (1697). (Opera I, 768-778.) 7. Bliss, G. A., Calculus of Variations. (Open Court Publishing Co., Chicago, 1925). 8. Bonnard, B., V. Jurdjevic, I. A. Kupka and G. Sallet, Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc. 271, no. 2, 525-535 (1982). 9. Bonnard, B., V. Jurdjevic, I. A. Kupka and G. Sallet, Systemes de champs de vecteurs transitifs sur les groupes de Lie semi-simples et leurs espaces homogenes (French). In Systems Analysis Conf. (Bordeaux, 1978), Asterisque, Soc. Math. France, Paris, 75-76 (1980). 10. Bonola, R., Non-Euclidean Geometry. (Italian original: La Geometria non-Euclidea, The Open Court Publishing Co., 1912.) Eng. transl. by H. S. Carslaw, (Dover, New York, 1955). 11. Cesari, L., Optimization—Theory and Applications. (Springer-Verlag, New York, 1983). 12. Clarke, F. H., The maximum principle under minimal hypotheses. SI AM J. Control Optim. 14, 1078-1091 (1976). 13. Clarke, F. H., Optimization and Nonsmooth Analysis. (Wiley Interscience, New York, 1983). 14. Clarke, F. H., Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory. Grad. Texts in Math. 178, (Springer-Verlag, Berlin, 1998). 15. Euler, L., Methodus inveniendi . . . accepti. Lausanne, Geneva, 1744; (Opera omnia, ser. I, vol. 25, 1952). 16. Goldstine, H. H., A History of the Calculus of Variations from the 17th to the 19th Century. (Springer-Verlag, New York, 1981). 17. Guillen, M., Five equations that changed the world: the power and poetry of mathematics. (New York, Hyperion, 1995).

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18. Hestenes, M. R., Elements of the calculus of variations. In Modern Mathematics for the Engineer, (E. F. Beckenbach Ed., McGraw-Hill, New York, 1956). 19. Isidori, A., Nonlinear Control Systems. (Springer-Verlag, Berlin, 1995). 20. Jakubczyk, B., and W. Respondek eds., Geometry of Feedback and Optimal Control. (Marcel Dekker, New York, 1997). 21. Jurdjevic, V., and I. A. K. Kupka, Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble) 31, no. 4, vi, 151-179. 22. Jurdjevic, V., and I. A. K. Kupka, Control systems subordinated to a group action: accessibility. J. Differential Equations 39, no. 2, 186-211 (1981). 23. Jurdjevic, V., and I. A. K. Kupka, Linear systems with degenerate quadratic costs. Compt. Rend. Math. Rep. Acad. Sci. Canada 10, no. 2, 101-106 (1988). 24. Jurdjevic, V. Geometric Control Theory. (Cambridge U. Press, New York, 1997). 25. Kupka, I. A. K., Degenerate linear systems with quadratic cost under finiteness assumptions. In New trends in nonlinear control theory (Nantes, 1988), Lecture Notes in Control and Inform. Sci. 122, (Springer, Berlin, 231-239, 1989). 26. Lee, E. B., and L. Markus, Foundations of Optimal Control Theory. (Wiley, New York, 1967). 27. Lochak, G., La Geometrisation de la Physique. (Flammarion, Paris, 1994). 28. Malisoff, M., On the Bellman equation for optimal control problems with exit times and unbounded cost functionals. In Proc. 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec. 1999, (IEEE Publications, New York, 23-28, 1999). 29. Malisoff, M., A remark on the Bellman equation for optimal control problems with exit times and noncoercing dynamics. In Proc. 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec. 1999, (IEEE Publications, New York, 877-881, 1999). 30. Nijmeijer, H., and A. van der Schaft, Nonlinear Dynamical Control Systems. (Springer-Verlag, New York, 1990). 31. Pontryagin, L. S., V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes. (Wiley, New York, 1962). 32. Saccheri, G., Euclides ab omni naevo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universale Geometriae Principia. (Milan,

165 Italy, 1733). 33. Stillwell, J., Sources of Hyperbolic Geometry. History of Mathematics, Vol. 10 (Amer. Math. Society, Providence, RI, 1996). 34. Struik, D. J., Ed., A Source Book in Mathematics, 1200-1800. (Harvard U. Press, Cambridge, Mass., 1969). 35. Sussmann, H. J., A strong version of the Lojasiewicz maximum principle. In Optimal Control of Differential Equations, N. H. Pavel Ed., (Marcel Dekker, New York, 293-309, 1994). 36. Sussmann, H. J., A strong version of the Maximum Principle under weak hypotheses. In Proc. 33rd IEEE Conference on Decision and Control, Orlando, FL, 1994, (IEEE publications, New York, 1950-1956, 1994). 37. Sussmann, H. J., An introduction to the coordinate-free maximum principle. In Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek Eds., (Marcel Dekker, New York, 463-557, 1997). 38. Sussmann, H. J., and J. C. Willems, 300 years of optimal control: from the brachystochrone to the maximum principle. (IEEE Control Systems Magazine, 32-44, June 1997). 39. Sussmann, H. J., Multidifferential calculus: chain rule, open mapping and transversal intersection theorems. In Optimal Control: Theory, Algorithms, and Applications, W. W. Hager and P. M. Pardalos Editors, (Kluwer Academic Publishers, 436-487, 1998). 40. Sussmann, H. J., Geometry and optimal control. In Mathematical Control Theory, J. Baillieul and J. C. Willems Eds., (Springer-Verlag, New York, 140-198, 1998). 41. Sussmann, H. J., Transversality conditions and a strong maximum principle for systems of differential inclusions. In Proc. 37th IEEE Conference on Decision and Control, Tampa, FL, Dec. 1998. (IEEE publications, New York, 1-6, 1998). 42. Sussmann, H. J., Resultats recents sur les courbes optimales. In Quelques aspects de la theorie du controle, Journee annuelle de la Societe Mathematique de France, (Paris, 1-50, June 2000). 43. Sussmann, H. J., A nonsmooth hybrid maximum principle. In Stability and Stabilization of Nonlinear Systems (Proceedings of the First European Community Nonlinear Control Network Workshop on Stability and Stabilization held in Ghent, Belgium, on March 15-16, 1999), D. Aeyels, F. Lamnabhi-Lagarrigue and A. J. van der Schaft Eds. Lect. Notes Control and Information Sciences 246, (SpringerVerlag, 325-354, 1999). 44. Sussmann, H. J., New theories of set-valued differentials and new versions of the maximum principle of optimal control theory. To appear in

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Part II Contributed Chapters

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169

SYMPLECTIC M E T H O D S FOR S T R O N G LOCAL OPTIMALITY I N T H E B A N G - B A N G CASE ANDREI A. AGRACHEV Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina ul.8 - 117966 Moscow, Russia and SISSA, Via Beirut 4 - 34014 Trieste, Italy E-mail: [email protected] GIANNA STEFANI Dip. di Matematica Applicata, G. Sansone, Via S. Marta 3 -50139 Firenze, Italy E-mail: [email protected] PIERLUIGI ZEZZA Dip. di Matematica per le Decisioni Via C. Lombroso 6/17 - 50134 Firenze, Italy E-mail:[email protected], URL: http://www.dmd.unifi.it/zezza/ We announce a new sufficient condition for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form. The controls appear linearly and take values in a polyhedron and the state space and the constraints are smooth, finite dimensional manifolds.

1

Introduction

We state a sufficient condition for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form. The controls appear linearly and they take values in a polyhedron and the state space and the constraints are smooth, finite dimensional manifolds; here we give only a sketch of the proof, a complete version of the result will be published elsewhere, see 7

The sufficient optimality condition contains two different sets of assumptions: the first (Assumption 1 and 2) concerns the regularity of the Hamiltonian flow while the second is the positivity of a suitable second variation. In the case of bang-bang extremals the control values belong to the boundary of the control set so that the kernel of the first variation does not contain any subspace and hence the usual second variation is not defined. To overcome this problem we consider the finite dimensional subproblem generated by perturbing the switching times. We prove that the sufficient second order

170

optimality conditions for this finite dimensional subproblem yield local strong optimality. We follow the Hamiltonian approach to strong optimality consisting in constructing a field of state extremals covering a neighborhood of the reference trajectory. This field of extremals is obtained by projecting, on the base manifold M, the flow of the maximized Hamiltonian emanating from the Lagrangian submanifold of the initial transversality conditions, which belongs to T*M; if this projection admits a Lipschitz continuous inverse, then we can estimate the variation of the cost at a neighboring trajectory by a function which depends only on the final point and it is hence independent on the control differential equation, the Lipschitz continuity of the inverse is guaranteed by the surjectivity of the tangent map. Let us point out that if the final point is fixed then the manifold of the admissible final points reduces to a singleton and hence, to obtain optimality, it is sufficient the surjectivity of the projection which corresponds to the classical non self-intersecting property of the state extremals, when the final point is not fixed, the optimality will also be guaranteed by the positivity of the second variation. The relations existing between the second variation and the symplectic properties of the Hamiltonian flow can be used to show that if the second variation is positive then the tangent map of the flow is surjective and the sufficient optimality conditions for the above mentioned function of the final point are satisfied. A crucial idea underlying the general approach is that the linear Hamiltonian flow of this linear quadratic problem has to be the tangent map of the flow of the maximized Hamiltonian. Some issues has already been addressed. In 5 we stated sufficient conditions for strong local optimality for an optimal control problem in Rn with unbounded controls, while in 4 we gave an intrinsic expression of the accessory problem and we studied the relations between the Hamiltonian flow and the index of the second variation. The geometric properties of the field of extremals needed to prove sufficient conditions for strong optimality has been studied in 6 . Similar conditions for time-optimality of a bang-bang trajectory have been obtained by A. Sarychev in 9 and 8 .

171

2

Main results

Let Xi, i = 1 , 2 , . . . , m be m distinct C°° vector fields defined on the C°° manifold M and let us define the multivalued function X:M^>TM,

X:x*+co{X1(x),...,Xm[x)},

where co denotes the convex hull. We consider an optimal control problem stated in the Mayer form Minimize

J(£) := co(£(0)) + cT{Z{T))

(2.1)

subject to i{t) e x{i(t)) ttO) 6 N0 , £(T) 6 JVr

(2-2) (2.3)

where N0, NT are C°° submanifolds of M. We are given, as a candidate optimal solution, a reference bang-bang Pontryagin extremal £; that is an absolutely continuous function £ : [0, T] - • M which is a solution of system (2.2)-(2.3), satisfies the Pontryagin maximum principle (PMP) and it is such that we can find a partition of the interval [0, T], {0 = t0 < tx < t2 < • • • < tr < U+i = T} with the property that, in each subinterval, the function £ is a solution of i(t) = XjMt)),

te[ti-i,n]

(2.4)

for some ji 6 { 1 , 2 , . . . , m } ; the values ti, i = 1,2,... ,r will be called switching times. We assume that Xj{ ^ Xji+1, for i = 1,2,..., r, for not introducing unnecessary switching times. Corresponding t o t h i s reference trajectory we define the time-dependent reference vector field h : [0, T] x M -> TM and we set hi ~ % 4 . _ l j t . ) : x >->•

Xj^x).

The time dependent reference vector field ht defines, by lifting to the cotangent bundle, the reference time dependent Hamiltonian H : [0,T] x T*M - • E, (t,£) ^

(£,ht(n£)),

where n : T*M -> M is the canonical projection and we set Hi := The maximized Hamiltonian H :T*M-*R,

l^

max

(l,v),

H^t._lit.y

172

is well defined and Lipschitz. For simpler notations we set x0 := |(0), xT := £(T). Recalling that any Hamiltonian Ht : T* M ->• R defines a Hamiltonian vector field, denoted by Ht, we can express the PMP by saying that there exist po £ {0,1} and a lift A of £ which is a solution of X(t) = Ht(X(t))

(2.5)

A(0) - po dco(x0) = 0 on TX0N0 \{T)+podcT(xT)=0 on TXTNT

(2.6) (2.7)

such that bo| + | | A | | ^ 0 Ht{X(t)) =

max (X(t),v) = H(X(t)). vEX(i(t))

The functions po CQ and po ex are denned on N0 and Nx, respectively, but they can be extended to the whole manifold M in such a way that the transversality conditions (2.6) and (2.7) hold on the whole tangent space. Let us denote by a, j3 : M —> E two functions such that a -p0c0 /3 — po ex

on N0, on Nx

A(0) = da(xo), X(T) =

-dP(xT).

In the normal case (po = 1) Q, /3 are cost functions equivalent to the original ones while in the abnormal case (po = 0) they are extensions of the zero function. When po = 0 all the costs disappear from our conditions and indeed we will study a problem with a zero cost. Proving that £ is a strict strong minimizer will imply that it is isolated among the admissible trajectories. The transversality conditions of the PMP can be stated saying that the end points of the adjoint covector belong to some specific Lagrangian manifolds, if we define A0 := {da(x) + T^No | x 6 N0} , A T := {-d/3(x) + T^Nx | x € NT} , then the transversality conditions (2.6)-(2.7) can be expressed as A(0) 6 A 0 ,

X(T) G A T .

173 The sufficient conditions will be stated analyzing the problem Minimize

J(fl := a(£(0)) + /?(£(T))

(P)

subject to (2.2) and (2.3). The points ti:=\(U),

i=0,l,...,r + l

will be called the switching points of the adjoint covector A. From the PMP we can deduce the following consequences which represent necessary optimality conditions for i — 1,2,..., r Hi(li) = Hi+i(£i) (d(Hi+1-Hi),Hi+1)(ei)>0. To state sufficient condition for f to be a strong local minimizer we need to strengthen these two conditions. We will assume that the following hold Assumption 1. (d(Hi+1 - Hi),Sw){ti)

> 0, * = 1,2,... ,r.

Assumption 2. The maximum max (\(t),v), t€[0,T] v€Xtf(t)) is attained at a vertex of X(£(t)) for all t ^ ti,t2,- • -,tr or along an onedimensional edge for t = t\, t2, • • •, tr. These conditions are strictly related to the properties of the flow of the maximized Hamiltonian H and they can be interpreted as regularity conditions on the maximized Hamiltonian near the adjoint covector. In fact the PMP implies that each switching point Hi belongs to the level set i?j+i — Hi = 0 and Assumption 1 yields that this level set is an hypersurface called switching surface. Moreover we can prove Lemma 2 . 1 . Under Assumption 1 there exists a neighborhood U of IQ such that we can define recursively, for i = 0 , 1 , . . . , r, the C°°-maps Ti : U - • M, and fa : U -> T*M

in the following way, set To : = 0,


the T{ 's are implicitly defined by (Hi - Hi+1) (expnWHiifa-iit))) Ti("i) = ti,

=0

174

while the fa 's are C°° symplectic diffeomorphism defined as <j>i{t) := exp(-Ti(e)Hi+1)

expn(e)Hi{4>i-i{l)).

Under Assumptions 1 and 2, by Lemma 2.1, we can say that the map "K : [0,T] x U - > [ 0 , r ] x T*M given by •K{t,t) = (t,exptHi+1(i(e))),

t £ [Ti(0,r i + i(/)],

is the unique solution of the Hamiltonian system \(t) = H(X(t)) A(0) = L Moreover the flow "K is C°° on each 2 n + 1 dimensional C°° manifold with boundary Oj, given by Oi := { ( M ) | / € U, Ti_i(0 < * < Ti(l)} C [0,T] x T*M. Let us remark that every solution of this system has the same number of switches as the adjoint covector A. Remark 2.2. Assumption 1 plays a role analogous to the one of the strengthened Legendre condition in the case of unbounded controls; in fact Lemma 2.1 says that we can define locally, in a tube around the adjoint covector, a timedependent Hamiltonian as {t,l)t-¥Hi(t,e),

for

M~1(t,t)£Oi.

Assumption 2, which concerns the uniqueness of the maximum point, ensures that with this choice we obtain the maximized Hamiltonian, for analogous conditions in the case of unbounded controls see 5 . In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial and hence the usual second variation, which is defined on the kernel of the first one, does not exist. To define an appropriate second variation we proceed starting from the following trivial fact: any choice of a subset of admissible variations for problem ( P ) would lead to a subproblem and hence to necessary conditions. Let us choose an appropriate r—dimensional family of variations corresponding to bang-bang trajectories, they are constructed by taking variations of the switching times. The optimality with respect to these variations will be sufficient (under the above stated assumptions) to prove strong local optimality of the reference trajectory. For a given a > 0 such that min i=l,...,r+l

(U — tj_i) > 2a,

175

let e G B(0, a) C W, set e 0 = £V+i = 0 and consider the time-dependent vector field obtained from the reference one in the following way (e,x) t-> ht(e,x) = hi(x) , t 6 (£,-_i + £•,•-!, *i +£i),

i = 1,2,.. .,r + 1.

In other words we have moved the switching time U by £j. Remark 2.3. Notice that a small e corresponds to a control variation which is small in the L1 norm but not in the L°° norm. We denote the flow of £(t) = ht(e,£(t)) by St : M x B ( 0 , a ) - » M and we consider the following finite dimensional subproblem of problem (P) Minimize

j(x,e)

:= a(x) + P(Sr(x,e))

(sub-P)

subject to x £ N0 , ST(x,e)

e NT.

Note that St : x ^

St(x,0)

is the flow of the reference vector field and hence St(xo, 0) = £(£)• Thanks to the choice of a,0 we have that the PMP implies that (xo,0) is a critical point for 7, that is dj(xo,0)=0, so that the second derivative of 7 at (xo,0) is well defined and J":=^27(xo,0) is a quadratic form on TXo M x E r which gives the second order approximation of 7. Thus the second variation of (sub-P) is the restriction of J " to the linearization of the constraints, namely if we set Ji = {{6x,e) e TXON0 x W : ST*{5x,e)

e

TXTNT},

then the second variation of (sub-P) is J,'^. Our main result is given by the following theorem. It states that under the regularity conditions on the maximized Hamiltonian, the positivity of the second variation of (sub-P) it is sufficient to prove that the reference trajectory is a strict strong local minimizer for the original problem. Theorem 2.4. Assume that a given bang-bang Pontryagin extremal £ satisfies Assumptions 1 and 2. If the second variation J!^ of (sub-P) is positive definite then £ is a strict strong local minimizer. In the abnormal case £ is an isolated admissible solution of the constrained control system.

176

3

Sketch of the proof

The proof is very technical, here we give an idea of how to construct the field of extremals and of the relations existing between the Hamiltonian flow and the second variation. A general introduction to symplectic methods and their applications to optimal control is in 1 and 2 . When the initial point is not free, it is not possible to cover, by the projection, a neighborhood of the initial point. In the calculus of variations this problem was solved by perturbing the initial time, but this method does not work in our case because the projection is singular on [£o,*i]- We can achieve the same result by perturbing the initial cost with a penalty term and constructing an equivalent problem with a free initial point. Let Q be any non negative quadratic form on TXoM, such that kerQ = TX0N0, we extend it to TXoM x E r by setting Q[6x,e}2 = Q[6x}2. If the quadratic form J" is positive on N then we can find p > 0 such that J" + PQ > 0 on {(Sx,e) G TXoM x W : ST*(Sx,e)

€ TXTNT} ,

as can be easily proved by elementary arguments of linear algebra. Let us choose a function ap such that ap = a on iVo dotp — da on TXo N0 D2(ap-a){xQ) = pQ, and consider the problem Minimize

a p (£(0)) + j8(£(T))

(3.1)

subject to

iit) G x(m) £(T) e NT. Since the reference trajectory satisfies the initial boundary conditions then proving that it is optimal for this new problem yields its optimality for the original one; therefore without loss of generality we can assume that the original problem has already the initial point free (iVo = M and a = ap) and hence the initial Lagrangian submanifold is horizontal and its projection covers a neighborhood of the initial point xoRemark 3 . 1 . This reduction is possible because the new cost on the initial point contains an exact penalty which can be constructed assuming that the second variation is positive definite.

177

Let s be the canonical one form in T*M, we denote by er = da the canonical symplectic two form on T*M, see 3 for the definitions. In the following we will use some of the properties of the Cartan form u> = s — H dt on I x T*M, associated to the Hamiltonian H. Namely • u> evaluated along a lift of a solution of (2.2) is non positive and it is zero along A. • w is exact on the manifold generated by the flow of H emanating from a Lagrangian manifold. By taking the restriction to a neighborhood of xo we can assume that A Q C U and that it is a smooth simply connected Lagrangian submanifold. We define r+1 4=1

The fi;'s are n + 1 dimensional C°°-manifolds with boundary. When the map -Kt := 7T o 0it : A 0 M- M

is nonsingular for all t € [0, T] then, by relating the properties of the Cartan form u> and those of this map, we can estimate the variation of the cost at a neighboring trajectory by a function which depends only on the final point and it is hence independent on the control differential equation. The optimality properties of | can be described through those of the symplectic map "Kt* '• Tf0Ao —> T^,t-.M, for simpler notations we set LQ := T(0 Ao- In particular let us consider, for i = 1,2,..., r + 1, the tangent map to the pull-back of the restriction of the flow "K to fli-i is given by the maps ^ : L 0 -»• TtoT*M ipi :=5ii.](exptiHi)tf(j)i-.it

=

ji^l^iexpti-iHi)*^-!*.

The functions ^,'s can be used to characterize the singularity of 7rt T h e o r e m 3.2. The map n* "Kt* • L0 -»• T^M is onto for each t £ [0,T], i.e.,

for each t 6 [0, T], if and only if

178

1. For i = 1,2,..., r -+-1 we have that ir*ipiL0 = TX0M. 2. For i = 1,2, ...,r

if Sli, Sl2 € L0 are such that 7r» tpi 5ti = 7T* rpi+iSl2

then adjide1,tpi+1S£2)>0. The above theorem states that, in the bang-bang case, a conjugate point can occur only at a switching time and, differently from the case of unbounded controls, the second variation may have a positive index without having positive nullity for a smaller t. When the final point is not fixed the following Theorem describes the optimality of £. Theorem 3.3. Assume that the map 7r» "Kt* '• LQ —> T;,t,M is onto for each t G [0, X1]. If for every non zero 6(.\ 6

LQ

7r» ipr+iS£i

and % £

^•T*^X(T)^T

SUC

^

^hat

= 7T» SI2

we have that o-(V»r+i«i, W2) > 0 then £ is a strict strong local minimizer for the problem (P). To show the relations between the second variation and the properties of the Hamiltonian flow, we first reduce (sub-P) to a problem which is linear in the control and where the admissible control maps are piecewise constant with the ti's as switching times. This goal can be achieved by a reparametrization of time. Consider the following control boundary value problem on E (p{r) = 1 + y>(0) = 0,

V(T)


where the control u takes values in (—1,1). Any solution of this boundary value problem is an increasing isomorphism of the interval [0, T] onto itself. If we set £i : = y - 1 ( * i ) ~U,

i=

l,2,...,r

179

then we have that ht(e,x) Moreover Sv^(x,e)

=

hv-i(t)(x).

is the solution of the differential equation C(T) = [1 + K T ) ] M C ( T ) ) ,

C(0)=X.

(3.2)

Let us notice that, since the behaviour of the trajectory is fully described by the integral of the control v, to obtain our variations it is sufficient to take v as a piecewise constant control with the same switching times of the reference trajectory. If i/; is the value of the control on [U-i, t{) and we set Ui ~

Vi (ti — i j - x )

then, from the definition and from the boundary conditions, we have that i

r+1

3=1

3=1

the last relation yields that our control space U has dimension r U:= ^ ( u i , u 2 , . . . , « r + i ) € # r + 1 | 5 T « i = o L For each u € U we denote by uu the corresponding control. In our finite dimensional subproblem (sub-P) we take the state defined by equation (3.2) and than the reference control will be the zero one. If we define the second variation of this problem, as in 4 , through the pull-back system, we obtain a linear-quadratic problem on TXoM x U. Thanks to the special structure of the problem the associated Hamiltonian is of the following type

(«,./)-• Gi'(MMt) where G" : T*QM x TXQM -> ffi is a piecewise constant linear Hamiltonian. We denote by Q"(6l,u) the solution of the associated Hamiltonian system \{t) = G'l{\{t))vu{t),

A(0) = St

The Lagrangian subspaces of the transversality conditions are L'J := {{-D2j(x0)(Sx,-),6x)

where 7 = a + f3 o St-

\ Sx G TXoM}

(3.3)

180

We study the changes of the signature of the second variation recursively, for k = 0 , 1 , . . . , r, on each

xk = xnTX0MnUk, where Uk := {u e U | Uj = 0 for j = k + 2 , . . .r + 1} , notice that UQ = {0} and Ur = U. This corresponds to consider admissible controls that are zero from a certain switching time on. Following the same ideas as in the case of unbounded controls, see 4 , we obtain that ker J!^ is described by those S£ G LQ and u G Uk such that %'^{5(.,u) £ L? and f G'i (S" (W,«)) i/„ (*) d* = 0,

for all veUk.

(3.4)

Thanks to Assumption 1, equation (3.4) defines a linear feedback laww : LQ —> U and S"(<W, (w, SI)) corresponds to the solution of the classical Jacobi system, since it is the solution of the Hamiltonian system of the second variation with the feedback control defined by equation (3.4). To conclude the proof we show that the linear maps

S$.(M):=S5r-(M,<w,M»), contain the necessary information about the behaviour of tangent map to the flow "K at the switching point £;. This information can be described through the anti-symplectic isomorphism i : T*oM x TX0M -> T(oT*M,

(w, 6x) •-> - u +

d(-0)t5x,

as S§. = »-Vfc+i * for * = 0 , 1 , . . . , r.

(3.5)

It is important to notice that (3.5) states that, modulo isomorphisms, the flow of the Hamiltonian of the second variation is the tangent map to the flow of the maximized Hamiltonian. References 1. Andrei A. Agrachev and Revaz V. Gamkrelidze, Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control, Hector J. Sussmann, Ed., Pure and Applied Mathematics, vol. 133 (Marcel Dekker, 1990).

181

2. Andrei A. Agrachev and Revaz V. Gamkrelidze, Symplectic methods for optimization and control, in Geometry of Feedback and Optimal Control, New York, New York, B. Jacubczyk and W. Respondek, Eds., Pure and Applied Mathematics, 1-58 (Marcel Dekker, 1997). 3. Vladimir I. Arnold, Mathematical methods in classical mechanics (Springer, New York, 1980). 4. Andrei A. Agrachev, Gianna Stefani and PierLuigi Zezza, An invariant second variation in optimal control, International Journal of Control, 7 1 , 5, 689-715 (1998). 5. Andrei A. Agrachev, Gianna Stefani and PierLuigi Zezza, Strong minima in optimal control, Proceeding of the Steklov Mathematical Institute, 220, 4-26 (1998); translation from Tr. Mat. Inst. Steklova 220, 8-22 (1998). 6. Andrei A. Agrachev, Gianna Stefani and PierLuigi Zezza, A Hamiltonian approach to strong minima in optimal control, in Differential Geometry and Control. Proceedings of the Summer Research Institute, Boulder, CO, USA, June 29-July 19, 1997, Providence, RI, Ferreyra et al., Eds., 11-22 (American Mathematical Society, 1999). 7. Andrei A. Agrachev, Gianna Stefani and PierLuigi Zezza, Strong optimality for a bang-bang trajectory, Tech. report 08 (Dipartimento di Matematica per le Decisioni, Firenze, 2000). 8. Andrei V. Sarychev, Sufficient optimality conditions for Pontryagin extremals, Syst. Control Lett. 19, 6, 451-460 (1992). 9. Andrei V. Sarychev, First and second order sufficient optimality conditions for bang-bang controls, SI AM J. Control Optimization, 1, 35, 315-340 (1997).

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183

C H A R G E S I N M A G N E T I C FIELDS A N D S U B - R I E M A N N I A N GEODESICS ALFONSO ANZALDO-MENESES AND FELIPE MONROY-PEREZ Departamento de Ciencias Bdsicas, Universidad Autonoma Metropolitana-azcapotzalco, 02200, Mexico D.F. E-mail: [email protected], [email protected] In this paper we study the action of certain nilpotent Lie groups under which a differential system given in terms of a rank-two distribution remains invariant. Such systems describe the dynamics of electric charges in static inhomogeneous magnetic fields. The action of the Lie group associated to the sub-Riemannian structures in the simplest non-uniform cases, contains naturally the Heisenberg group which corresponds to a uniform field. A linear magnetic field yields the nilpotent group associated with the so-called Martinet case and the group corresponding to systems of Engel type. The Cartan five dimensional system is also the natural setting for the study of linear magnetic fields acting on classical electric charges. Our results generalize previous works on this class of non-holonomic systems.

1

Introduction

Let A be a rank-two distribution denned on a differential manifold. We study the action of certain nilpotent Lie groups that leave the differential system determined by A invariant. The motivation for this study is the geometry of dynamical systems describing the motion of classical electric charges in the Euclidean plane, under the influence of external, perpendicular and inhomogeneous magnetic fields. Sub-Riemannian geometry provides a natural setting for the study of certain aspects in classical particle and plasma physics 5 , this approach is not new and has been in the literature at least in the last ten years, see for instance, 13 . The dynamics of a classical electric charge in the plane under the influence of a perpendicular inhomogeneous magnetic field can be described by means of a rank-two distribution, which together with a flat Riemannian metric, determines a sub-Riemannian structure. The Lie algebra generated by the distribution turns out to be nilpotent. The study of the action of the corresponding nilpotent Lie group plays an important role in our analysis, and it contains as the simplest nilpotent subgroup the well-known Heisenberg group. The so-called Martinet case, studied by R. Montgomery, 14 , and B. Bonnard et al, x, turns out, under our approach, to be a particular example

184

corresponding to linear magnetic fields. Our work is related with that of Y.L. Sachkov 17 and that of the two authors mentioned above. In contrast with those references, we present explicit and detailed calculations of trajectories (sub-Riemannian geodesies), and a more general approach. The PfafHan system associated to the inhomogeneous magnetic field leads us to consider Goursat and Cartan distributions. The non-holonomic constraints encoded in the set of 1—forms, motivates the extension of the standard two dimensional Lagrangian to higher dimensions, even in this generality, certain constants of motion are explicitly exhibited. For this presentation we apply variational principles and perform an explicit integration process in terms of hyper-elliptic functions. Abnormal extremals appear, however they turn out to be non strictly abnormal. In a separate work 12 , we apply Pontryagin Maximum Principle to carry out the study of a time optimal control problem in certain nilpotent Lie groups which contain some of the groups described here. Apart from this introduction, this paper contains four sections. In the second section we establish our notation, and provide basic definitions and standard results in sub-Riemannian geometry and distributions. In section three we apply the Lagrange multipliers technique and explicitly integrate the extremal equations in terms hyper-elliptic integrals. In section four, we discuss low dimensional cases. In particular we study the five dimensional Cartan system, which has attracted a lot of interest in differential geometry and control theory, see 3 . Also in this case we present an explicit integration in terms of the Weierstrass p-function. 2

Sub-Riemannian geometry and classical particles

As we mentioned before, some problems in classical particle physics find a natural setting within a general geometric framework which goes in the literature under different names: singular Riemannian geometry 2 , sub-Riemannian geometry 19 , or Carnot-Caratheodory geometry 7 . 2.1

Sub-Riemannian geodesies and distributions

A sub-Riemannian structure on a manifold P consists of a couple (A, g), where A is a distribution satisfying the so-called full-rank condition (Lie(A)(g) = TqP, for all q € P), and g is a Riemannian metric on A, see 8 . Let us denote the corresponding scalar product as (-,-) 9 . An admissible curve for A, also called horizontal, is an absolutely continuous curve

185

q : [0, Tg] -> P such that q(t) £ A(g(£)) almost everywhere. The length of an admissible curve q(t) is given as

L{q{t)) = J

9

^J(q(t),q(t))g dt,

whereas its energy is given as follows

E(q{t)) = \ j '(q(t),q(t))g dt. Given two points qo,qi 6 P, a minimizer connecting qo with q\ is an admissible curve q : [0, Tq] —> P with minimal length satisfying q(0) = qo and q(Tq) = qi. This defines the sub-Riemannian distance between qo and q\. It is well known that if all the curves considered are defined in the same interval, then the length and energy minimizing problems are equivalent. By the Chow-Rashevsky's theorem 6 , 14 , if the distribution satisfies the full-rank condition, and the manifold is connected, then any two points can be connected by a smooth horizontal path. A non-constant admissible curve q : [0,Tq] —¥ P is called a subRiemannian geodesic, if its restriction to any small enough subinterval of [0, Tq] is a minimizer, and ||<7(£)||9 does not depend on the parameter t. In contrast with the Riemannian case, sub-Riemannian geodesies are not determined by its initial tangent vector, but by its initial covector, that is, if p 6 T*P is an initial covector, then the corresponding initial tangent vector v is denned by means of the following formula (w,v)g

=p{w),

for all w S A(TT(P)), where IT denotes the canonical projection of the cotangent bundle T*P. We shall consider in this paper bracket generating and regular distributions. A distribution A on a manifold P is said to be bracJcet generating, if there is an integer n such that A™ = TqP for each q € P. Here, A i + 1 = [A,A 4 ], i = l,..., which lead to the flag A , c A^ c • • • C TqP

186

As R. Montgomery has pointed o u t 1 5 , the growth vector is the most basic numerical invariant associated with a distribution. The growth vector nq of A at the point q, is defined as nq = (niq,n2q,...

,nnq),

where n ^ = dim Aq. The distribution is said to be regular at q, if the growth vector is constant in a neighborhood of the point q. The notion of symmetry in sub-Riemannian geometry has been studied since the early paper of R. Brockett 2 . More recently symmetries have been analyzed for higher dimensional Heisenberg groups n , and for some relevant cases of low dimensional Lie algebras 17 . We shall follow our paper mentioned above to describe the symmetries of A by means of the action on P of a Lie group under which A remains invariant. 2.2

Geometry of non-holonomic

constraints

The dynamics of a point under non-holonomic constraints when these are given by a certain Pfaffian system, can be written as a differential control system. The corresponding variational problem of minimizing the action, can be formulated either as an optimal control problem, or in the language of connections on principal bundles. Let M be the base manifold, a simply connected and complete twodimensional Riemannian manifold and take a coordinate chart with coordinates (x,y). Let {d6i — {vidx + fiidy)}i be a set of linearly independent 1—forms, the kernel of this Pfaffian system is encoded in the differential control system q = uX1{q)

+vX2{q),

where the control parameters u = x, v — y and the state variable q = (x,y,6\,...) are taken in the appropriate spaces denoted as U and P respectively. If A = {Xi,X2} turns out to be a regular and bracket generating distribution, then for all q € P, we can complete a basis {Xi(q),X2(q),X3(q),... ,XdimP(Q)} for each tangent space TqP. By declaring this basis orthonormal we obtain an inner product (-, -)q whose restriction to A yields a sub-Riemannian structure. The sub-Riemannian geodesies correspond to absolutely continuous solution curves [0, Tq] ->• P, t H» q(t) of the above control system which minimize the energy functional

187

E(q(t)) = \jq<«(*),«(*)>, dt=^J

\2+v2

dt.

In summary, we obtain under this setting an optimal control problem on the manifold P. The above formulation leads us naturally to the framework of principal bundles. Because in our case (not true in general), the Lie group G, corresponding to the Lie algebra go generated by the vector fields {X3,... , Xdim p } , yields an action • : G x P -> P satisfying {9192)-q = 9i- (92 • q) for each q € P, which together with the natural projection TT : P -> M, q H* Tr(q) = (x, y), determines the principal bundle over M with total space P and structure group G. Furthermore, since for each q 6 P one has the splitting TqP = Horq © Verq, where Horq = Aq and Verq = go(q), then a unique connection on the principal bundle (P, M, n, G, •) is determined, see 18 for details. Remark 2.1. The control system mentioned before determines horizontal lifts Xi and X2 of tangent vectors on M. This lifting provides a smoothly varying family of linear maps aq : T(x>y)M —*TqP satisfying dirq o aq — idM and a(g-q) = g- a{q) 2.3

The variational Problem

As explained above, the physical problem of a classical particle evolving on the manifold M under the influence of an external static magnetic field normal to M, can be formulated indistinctly either as a variational problem, as an optimal control problem or as a problem of connections on principal bundles. In this paper we take the calculus of variations' approach to discuss the problem. Let us choose the coordinate chart (x, y) on M in such a way that the magnetic field B is written locally as (0,0, Bz). We shall consider two cases, namely Polynomial magnetic fields The function Bz (x) is a polynomial in x Linear magnetic fields The function Bz(x,y)

is linear in both x and y

If m denotes the particle's mass, the geometrical problem consists in the calculation of the extremum of the action

S=jJ(x>+y>)dt,

188 subject to nonholonomic constraints which correspond to particular gauge choices for the magnetic field vector potential. 3

Polynomial Magnetic Fields

Several choices are possible to define a polynomial Bz(x), in particular we consider nonholonomic constraints defined by means of the following Pfaffian system ojk=d9k-l-xkdy,

k = l,...,N

+ l,

(3.1)

where N is the degree of Bz(x). These 1—forms have a well defined degree. Setting deg(x) = deg(y) = 1, we get deg(uik) = k + 1. Particular cases for which some of these forms are absent could be of some interest. This variational problem lead us to a rank-two distribution A defined on the N + 3-dimensional manifold P, with coordinates q = (x,y, 6\,... , #;v+i), as explained before. The distribution A is given by the vector fields

Xi = dx, N+l

X2 = dy+Y^

-^dOi.

(3.2)

This distribution is bracket generating and its growth vector is generically written as ( 2 , . . . , N + 3); the last entry can be smaller under some circumstances. The Lie brackets lead to a nilpotent Lie algebra denoted as Q, with the following commuting relations [Xi,X2] = X3, [Xi,X3] = X4,

[XI,XN+2]

with

= Xtf+3,

189

N+l

1

The vector field XN+3 — 86N+I is central, and the subalgebras generated by the vector fields {Xk, k ^ 1} are Abelian. The subalgebra generated by {Xi,X3,Xi,... ,XN+3} is a non-Abelian ideal, whereas those generated by {Xi,Xi+\,... ,XN+3} are Abelian ideals. Let * be the product on P defined as ( a i , a 2 , . . . ,aN+3)*(a[,a'2,...

,a'N+3)

= (a'{,a2',...

,"#+3)

where a" = a i + a[, "2 = a 2 + a 2 ,

fc-i

ai' = a*+aj;+ V 71

1

^a?~X>

k = 3,...,N + 3

The manifold P becomes a Lie group with this product and we have Theorem 3.1. The group (P, *) acts on itself by left multiplication leaving the vector fields {X\,..., Xw+3} invariant Proof. It follows directly from the definition of *-left invariant vector fields.

• From the physical point of view we can write the following Corollary 3.2. The dynamical system defined by the metric x2 + y2 and the non-holonomic constraints u>i = dBi — \xldy, i = 1 , . . . , N + 1 is invariant under the left action of P. Remark 3.3. Observe that the vector fields {X{} are not left invariant with respect to left translations of the group law locally obtained by the exponential map, that is a" = ai + a[,

a'2' = a2 + o>'2, a'l = a3 + a'3 + -(ctict'2 — 0.^2), • • • ,etc.

190 However, any linearly independent set of vector fields left invariant with respect to this group law, can be expressed as linear combination of the basis {Xi} above. These linear combinations determine a map from the tangent space to itself and are directly related to the gauge invariance of the magnetic field vector potential. Clearly, the set of all linear maps of the corresponding vector space forms a group. Accordingly, there is set of related Pfaffian systems corresponding to the same dynamical system. We are dealing here with the group of automorphisms Aut(g) of our nilpotent Lie algebra g.

3.1

Goursat systems

The Goursat normal form for a Pfaffian system has been in the control theory literature under the name of chained form systems. This normal form has been used in the study of planning problems with non-integrable velocity constraints, see 16 . A Pfaffian system on a N + 3 dimensional manifold P system is in the Goursat normal form, if there exist a coordinate chart q = (x,y,ui,... ,UN+I) on which the system is written as follows

du\ — ydx du2 — u\dx \

(3.3)

dv,N+l ~ Ujydx

This system, together with a flat Riemannian metric on P, is equivalent the system discussed above, in the sense that it leads to the same equations of motion and it is possible to give an analytic map from one set of variables to the other. We have the following Proposition 3.4. The Lie algebra determined by system (3.3) is isomorphic to the Lie algebra g. Proof. In fact, the kernel of the above Pfaffian system is encoded in the following differential system

q=

xZ1(q)+yZ2(q),

defined by the following vector fields

191 N

Z\ = dx + ydui + 22 Uidui+x t=i

Z2=dy The non-zero Lie brackets are the following [Z1,Z2] =

-du1=:Z3

[ZuZk]

= (-l)* + 1 0u f c _i =: Zk+1

[Z1:ZN]

= (-l)N+1duN

=: ZN+1

Therefore, the Lie algebra coincides with Q above as expected. 3.2

(3.4)

•

Sub-Riemannian geodesies for polynomial fields

In this section we shall integrate the extremal equations. This equations are written in the total space P, and project down to the base manifold M. That is, consider a curve C : [0, Tc] -+ M, t »-> C(t), with initial point {xo,Vo) = C(0) and with p0 in the fiber 7r -1 (xo,yo). The parallel transport of po along C is given by the curve defined by dC{t)/dt = apdC(t)/dt, with (7(0) = po and C(t) = p. The curve C projects by 7r onto the curve C. As usual in optimal control problems, there are two types of extremal curves, normal and abnormal, in the present context the latter correspond to those extremals which do not depend on the kinetic energy. For details on abnormality we refer the reader to R. Montgomery's survey 14 . Proposition 3.5. The normal extremals project to the curve t i-> C(t) = (x(t),y(t)) which satisfies the system x = jj =

qBz(x)y -qBz{x)x

with qBz(x) a polynomial in x. And consequently can be integrated by quadratures. Proof. For the Pfaffian system (3.1) and the metric given by the kinetic energy, the Lagrangian is given by

192

C = A 0 y ( x 2 + y2) + j ; \k0k

-

-xky),

k=i

where the A*, A; = 1,...,JV + 1, are Lagrange parameters, in general time dependent, and the parameter Ao is a constant, which can be taken equal to 1. The Euler Lagrange equations readily yield Aj

Et JTWX

mx = my =

. (i-iyx

j_i .

y

'

x

'

Ai = 0, consequently, all Lagrange parameters A; are constant. The above equations are just Lorentz' equations for a particle charge q with mass m, under the influence of the magnetic field given by B = (0,0, Bz)1 and qBz = — ]T^ (i-\)\x%~1 •> which i s a polynomial of degree N in x. The horizontal trajectory C(t) is given by the solutions x(t) and y(t) and by the solutions of the Pfaffian system 0* — \xly = 0, for i = 1 , . . . , N + 1 , as required. In order to integrate the equations we parameterize the trajectories by arclength, write the field as ^qBz = ^ bix1, and set, without loss of generality, the coordinate chart in such a way that (xo,yo) is the origin in the plane, we have then N

bixly,

x = ^2 i=0

d

TT-^

bi

i+1

«=0

so that integrating N

h °i

E TT\ i=0

and then

X

i+1

'

193 h a2N+2X27V+2

£ = a0 + a\x +

with £ = x and a set of known constants a i5 i = 1 , . . . , 2N + 3. Some of these constants are the following

a0 = (iof, ai = 26 0 «/o, a-2 = hy0 - b%,

2bNbM-i

&2N+1

a2N+2

N(N + 1) _

b%

The above equation can be formally integrated by means of the following hyper-elliptic integral

t(x) =

/ (0) y/a0 H

h a2N+2C2N+2

'

the associated inverse functions are thus the solutions of the above algebraic equation and the integration is complete.

•

Remark 3.6. The coefficients Oj can lead to singular algebraic curves for certain initial conditions. For example, for XQ = 0 and bo = 0, i.e., ao = 0 = a\, the curve has at least a double point for N > 0. Since qBz{x) = bo + b\x + • • •, this example corresponds to a trajectory along a line (the yaxis) for which the magnetic field is zero. Since in this case the remaining parallel lines on which the magnetic field is zero can be at any other values of x, then the same applies for all straight lines on which B is zero. We now discuss abnormal extremals for polynomial fields. We consider in this case the Goursat normal form for the Pfaffian system which is equivalent to our system as was mentioned above. We take Liu and Sussmann's definition of strict abnormality, see 10 Definition 3.7. A strictly abnormal extremal of a sub-Riemannian manifold, is an abnormal extremal that is not normal.

194

Proposition 3.8. Abnormal extremals for polynomial magnetic fields in one variable correspond to straight lines on M, and consequently they are not strictly abnormal. Proof. In this case we have the Lagrangian JV+l

£ abnormal G

\

"* \ / -

=2-,

Afc Ufc

'

„.

*\

~uk+lx),

i=l

and the Euler-Lagrange equations reduce to qBz(x)y

= 0,

qBz(x)x

= 0,

where again B = (0,0, BZY and qBz = \N+I, with constants of integration a. The non-trivial solutions are precisely the set of points for which Bz(x) = 0, which corresponds to straight lines, i.e., x = y •=• 0, in base space. Therefore, the abnormal extremals are not strictly abnormal. D Remark 3.9. We observe that, the equations of motion resulting from other gauges are equivalent to these obtained equations above. Furthermore, the extremal for the case n = 1 corresponds to the famous example of R. Montgomery and I. Kupka, see 9 . 3.3

Engel and Martinet systems

There are two particular cases which has been extensively studied by Y. Sachkov 17 , R. Montgomery 14 and B.Bonnard et al x , they are known as Engel and Martinet systems. Engel case In this case one considers the polynomial gauge for N = 1 and writes z = d\ and w = 82 in (3.2). The resulting vector fields are the following

195

Xi = dx, x2 X2 = dy + xdz + —dw, X3 = dz + xdw, Xi = dw, these vector fields yield a basis for the so-called Engel algebra

[X\, X2] = Xz,

[Xi, Xz] = Xi.

This algebra is four dimensional and the associated one-forms are of Goursat type. The growth vector of the distribution {X\,X2} is (2,3,4). The corresponding magnetic field is ^-Bz = b^x + boMartinet case In this case, one takes ^Bz = bix, that is, zero constant magnetic field contribution. From the two 1—forms of the Goursat system resulting for N = 1 in (3.1) only that corresponding to the single Martinet 1—form dw = dz — \x2dy is considered. The growth vector is here (2,3) for x 7^ 0 and (2,2,3) for x = 0. Notice that the reduced vector fields still satisfy the Engel algebra. Remark 3.10. R.Montgomery uses also the approach of classical particles in magnetic fields to describe the Martinet case, see 1 4 . 4

Linear magnetic fields, and Cartan's five dimensional case

We study now the system corresponding to Bz(x,y) being a linear function in the two variables x and y. This apparently simple case, leads naturally to the famous Cartan five dimensional Pfaffian system 4 . According to the last section, the points on which the magnetic field is zero are always straight lines parallel to the y—axis. Such a selection is in fact particularly simple, however, the generic case would be straight lines which are not necessarily parallel to the vertical axis, but have a nontrivial slope. For this purpose we take the function ^Bz with a2 + P2 = 1.

= b1(ax + l3y) + bo,

196

This case does not reduce to the case TV = 1 of the Goursat system (3.3) d>i = du — zdx u>2 = dz — ydx,

(4.1)

since we must add the following 1-form

Q3 = dv - -zdy.

(4.2)

The vector fields of the rank-two distribution denoted as V, corresponding to these forms are the following Xi — dx + zdu + ydz X2 = dy+

Z

-dv.

(4.3)

Let us denote as ns the nilpotent five dimensional Lie algebra generated by T>, the non-trivial Lie brackets are as follows [xltX2]

=^dv-dz=:X3

[x1,X3]=du=:Xi

(4.4)

[x2,X3] =8v=:X5. The growth vector of T> is (2,3,5). This system is usually called a system of Cartan type, it has been largely studied beginning with E. Cartan 4 , who showed their relation with the exceptional Lie algebra Q2- A modern treatment can be found for instance in 3 . The Lie algebra ns is more general than the algebra (3.4) corresponding to N = 1, since it contains the additional (central) generator X&. Let us denote as N5 Lie group corresponding to ns. This group can be represented by M5 endowed with the following group law

(ai,a 2 ,a3,0!4,a:5)(/3i,/?2,/33,/?4,/?5) = («i + / ? i , a 2 + /32,a'3,a'4,a'5), here

197 a3 = a3 + fa + - ( Q I / ? 2 -

a2Pi)

a'i = a 4 + Pi + 2("i^3 - a3Pi) - — (/?i - a i ) ( a 1 ^ 2 - a 2 f t ) a^ = a 5 + ^5 + - ( a ^ - 0:3^2) - y ^ C ^ ~ a2)(ai/S 2 -

a20i).

The group JV5 acts on itself by left multiplication leaving the following two vector fields invariant

x,=fc-|«,-(j| + i)a.-£*,. These vector fields span the kernel of the Pfaffian system wi = dz + %(ydx — xdy), Z

X

u2 = du + -dx + jz(ydx Z

- xdy),

(4.5)

1J

u3 = dv + -dy + jziydx

-

xdy),

that we call the Cartan Pfaffian system. A direct calculation shows that

[xl,x2] = dz + ^du + y-dv.= x3. Therefore taking X4 := X4 and X5 := X5, we recover the commuting relations (4.4) for the vector fields Xi, i = 1 , . . . , 5. In summary we have the following Proposition 4.1. The group N$ acts on itself by left multiplication leaving the vector fields Xi,i = 1 , . . . , 5 invariant. Moreover the Lie algebra generated by {Xi) is isomorphic to ns 4-1

The action of the semi-direct product N$ x S 0 2

As in the Heisenberg case there is a rotational action on N5, which is encoded in the left invariant vector field W = xdy — ydx + udv — vdu,

198 which satisfies the following commuting relations [XUW}= X2, [X2,W] = -X1, [X4,W}= X5, [X5,W]=-Xi, these brackets together with (4.4), allow us to consider the semi-direct sum ns ® s so 2 , from which the action of iV5 x SO2 becomes transparent. As usual, we take the product (x, z, w, R) • (x1, z',, w', R!) to be equal to (x+Rx*,z + z'+^x*JRxH,w with R,R' e S 0 2 ,

+ Rw' + hz'x-zRxH)

&-{x,y),

uf = (u,v),

+

^x4JRx'(x-R&),RR')

z £ E, and

We have the following Theorem 4.2. The group (iV5 x SO2, •) acts on N$ leaving the Cartan Pfaffian system (4-5) invariant. 4-2

Sub-Riemannian geodesies for linear magnetic field

As in the last section we take a curve C : [0, Tc] -> E 2 , t i-> C(t), with initial point (a;0, yo) = C(0) and with p0 in the fiber 7r_1 (XO, yo)- The parallel transport determines the curve C which projects by it onto C. Proposition 4.3. The normal extremals project to the curve t t-^ C(t) — (x(t),y(t)) which satisfies the system x =

m

—Bz(x,y)y

V=

Bz(x,y)x m with qBz(x,y) linear in x andy. The complete system in E 5 can be integrated by quadratures. Furthermore there are not strictly abnormal extremals. Proof. The Lagrangian for the non-holonomic constraints in this case is given as follows

199

Cc = A o y ( i 2 + f) + ^(u+-x

+

+ Ai(i + - ( i / i - xy)) ^x-^)

+

X3(v+-y

+

—

--y),

we take first the normal case, Ao = 1, to obtain easily x =

—Bz{x,y)y

V

Bz(x,y)x m with ^-Bz = bi(ax +/3y) + bo as required. Furthermore, the remaining EulerLagrange equations yield

z .

x .

z .

y

This system can be written in a simpler way by means of the following rotations

" P\ fx\ -fx\

( a 0\ [A

-fu\

-/3a)\yJ-\Y)'

\-/3aj{vJ

\V J

to obtain X = {hX + b0)Y, Y = -{hX + b0)X, Z=±(XY-YX),

Z V

Y

•

(4.6)

Y +

•

= -2 -6ZThe equation for X is just an elementary elliptic curve and its solution can be easily given in terms of doubly periodic functions. The corresponding algebraic equation takes the form

200

£2 = a 0 + a i X + a2X2 + a 3 X 3 + a 4 X 4 , here ao = X$, a\ = 2boYo, a2 = biYo — b\, a 3 = — &0&1 and 04 = —b\/A. The algebraic equation can be given in terms of the Weierstrass p-function by a linear fractional transformation X =

ap + b ;, cp + d

ad — be = 1,

where the p-function satisfies (p') 2 = 4 ( P - ei)(p - e 2 )(p - e 3 ) for ei, e 2 and e 3 constants. The function y ( i ) can be also obtained in terms of elliptic functions as follows

= Y0t- f (bo + h—^T.

2 the integration can be performed again analytically. For the old variables (x,y), the straight lines for which ^Bz = b\{ax + 0y) + bo vanishes are particular solutions of the equations of motion for the normal extremals. Since for the abnormal extremals, Ao = 0, the same straight line solutions are obtained, we conclude that also in this case the abnormal extremals are not strictly abnormal. Further, every horizontal vector is tangent to an abnormal extremal for this distribution.

•

Remark 4.4. Not surprisingly, in the coordinates (x,y,z,u) the system (4-6) corresponds to the case of an original Engel structure. In general, if the magnetic field does not variates strongly as a function of the coordinates, this case corresponds to the first order approximation. Remark 4.5. A simple characterization of the above elliptic curve is obtained in terms of the cross-ratio

ei - e 3 Most important is the fact that the modular invariant

201

tf_ (A2 - A + l ) 3 32 ' A 2 ( A - 1 ) 2 ' is a true modulus, that is, it can be used to define an equivalence between curves having the same J. If e? — ez,e\ = e-i or e\ — e$, the cross-ratio takes the values (0,1, oo) and J = oo. The curve degenerates in equivalent stable singular cubic curves. Topologically, these curves are associated with a sphere with two points identified. In our problem, if we let b\ —> 0 the curve degenerates in a rational curve. Acknowledgements Part of this work was done during a sabbatical stay of the first Author at the Arnold Sommerfeld Institute, Clausthal, Germany with support of DAAD. References 1. Bonnard B., Chyba M. and Kupka I., Lieu conjugue en geometrie sousRiemannienne. Le cas Martinet plat, Rapport de Recherche No.90, University de Bourgogne, Laboratoire de Topologie, Dijon, Prance, (1996). 2. Brockett R.W., Control theory and singular Riemannian geometry, in New directions in applied mathematics, Hilton, P.J. and Young G.S. Eds., (Springer-Verlag, 11-27, 1981). 3. Bryant R., Hsu L., Rigidity of Integral Curves of Rank-two distributions, Invent. Math., 114, 435-461 (1993) 4. Cartan E., Les systemes de Pfaff a cinque variables et les equations aux derivees partielles du second ordre, Ann. Sci. Ecole Normale, 27, No.3, 109-192 (1910). 5. Chandrasekhar, S., Plasma Physics, (The University of Chicago Press, Chicago, 1960). 6. Chow, W.L., Uber Systeme von Linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117, 98-105 (1939). 7. Gromov M., Carnot-Caratheodory spaces seen from within, in SubRiemannian geometry, Eds. A. Bellaiche, J.J. Risler, (Birkhauser, 79323, 1996). 8. Kupka I., Geometrie sous-Riemanniene, Seminaire Bourbaki, (1996). 9. Kupka I., Abnormal extremals, preprint, (1992). 10. Liu W., Sussmann H., Shortest paths for sub-Riemannian metrics on rank-two distributions, Memoirs of the American Mathematical Society,

202

118, No.564, (1995). 11. Monroy-Perez F. and Anzaldo-Meneses A., Optimal control on the Heisenberg group, Jour. Dyn. Control. Syst, 5, No.4, 473-499 (1999). 12. Monroy-Perez F. and Anzaldo-Meneses A., Optimal control on certain nilpotent groups, preprint, (1999). 13. Montgomery, R., The Isoholonomic Problem and some Applications, Comm. Math. Phys., 128, 565-592 (1990). 14. Montgomery, R., A Survey of Singular Curves in Sub-Riemannian Geometry, Jour. Dyn. Control. Syst, 1, 49-90 (1995). 15. Montgomery, R., A tour of Sub-Riemannian Geometries, Geodesies and Applications, to be published, (2000). 16. Tilbury D., Murray R., Sastry S., Trajectory Generation for the N-trailer Problem Using Goursat Normal Form, IEEE Transactions on Automatic Control, 40, 5, 802-819 (1995). 17. Y.L. Sachkov, Symmetries of Flat Rank Two Distributions and SubRiemannian Structures, Rapport de Recherche No.151, Universite de Bourgogne, Laboratoire de Topologie, Dijon, France, (1998). 18. Spivak M., A comprehensive Introduction to Differential Geometry, Vol. II, (Publish or Perish, 1979). 19. Strichartz R.S., Sub-Riemannian Geometry, Journal of Differential Geometry, 24, 221-263 (1986).

203

TOPOLOGICAL V E R S U S S M O O T H LINEARIZATION OF CONTROL SYSTEMS LAURENT BARATCHART INRIA, B.P. 93, 06902 Sophia Antipolis cedex, France E-mail: Laurent. [email protected]

Dept. of Mathematics,

MONIQUE CHYBA Univ. of California, Santa Cruz, CA 95064, U. S. A. E-mail: [email protected]

JEAN-BAPTISTE POMET INRIA, B.P. 93, 06902 Sophia Antipolis cedex, France E-mail: [email protected] This note deals with "Grobman-Hartman like" theorems for control systems (or in other words under-determined systems of ordinary differential equations). The main results (proved elsewhere) is that when a control system is topologically conjugate to a linear controllable one, then it is also "almost" differentiably conjugate. We focus on the meaning of this result, and on an open question resulting from it.

1

Introduction

In this note, we discuss the local behavior of a nonlinear control system x = f(x,u),

xeW1

, u£Rm

,

(1.1)

say around (0,0) € E n + m . For general control systems (as opposed e.g. to affine in the control), "local" has to be understood with respect to both state and control. The first reaction when dealing with local properties is to compute the linear approximation of (1.1). When this linear control system happens to be controllable, all the local usual control objectives can be met using linear control, based on the linear approximation. For instance, a linear control that asymptotically stabilizes the linear approximation will also stabilize the nonlinear system, locally; minimizing a quadratic cost can also be achieved up to first order based on the linear approximation only. Hence, the linear approximation is a good enough model for the purpose of designing controllers achieving a desired behavior for small states and controls. We believe that all control engineers or control theorists agree on this statement, arising from practice, although we would welcome some contradiction.

204

Rephrasing the above statement without reference to control objectives leads to an imprecise statement, grounded mostly on some necessarily subjective intuition, and that should rather be taken as an opening sentence to launch a debate than as a conjecture : (nothing distinguishes qualitatively the behavior of a nonlinear control ]system from the one of its linear approximation if the latter is controllable. (1.2) It is natural to try to formalize this statement, as a prerequisite to any proper theory of nonlinear modeling and identification of control systems, in a very preliminary manner since it only deals with local phenomena. A nice way to turn that belief into a sound, and correct, assertion would be to find some equivalence relation between control systems (or models) that preserves at least "qualitative" behavior, and for which these two systems (a nonlinear system and its controllable linear approximation) are in general equivalent. We assume controllability of the linear approximation because when this fails none of the above is correct, at least in most common case when the nonlinear system is itself controllable. Indeed, (non-)controllability is a qualitative phenomenon : for instance, feeding a linear non controllable system with "random" inputs, one observes that the state is confined in leafs of positive codimension, while for a controllable system the whole state space is explored.. To enlighten the discussion on local behavior of control systems, let us recall the situation for ordinary differential equations x = F(x) (particular case of (1.1) where the control u has dimension 0) : • If F(0) ^ 0, the "flow-box theorem" (see e.g. 1 ) , gives local coordinates,

(o\ smooth if F is smooth, in which F is of the form

:

w

• If F(Q) = 0 and the square matrix F'(0) has no pure imaginary eigenvalue (hyperbolic equilibrium), then Grobman-Hartman Theorem 5 tells us that the flow of the differential equation is locally conjugate to the flow of its linear approximation via a homeomorphism that need not, in general, be smooth if F is smooth (and in fact smooth conjugation requires more assumption, resonances are obstructions to it) (see e.g. 2 ) . • If -F(O) = 0 and the square matrix -F'(O) has some pure imaginary eigenvalue, then the situation is more intricate even locally, namely the phase

205

portrait of the nonlinear dynamical system x = F(x) can be very different locally from the one of a linear system. This case is of high interest in the theory of dynamical systems, but can be considered as "degenerate", in the same way as non controllability of the linear approximation for control systems. Since conjugation of flows does preserve qualitative phenomena like the overall aspect of the phase portrait, one can indeed assert that, locally around all points except non hyperbolic equilibria, a differentiable dynamical system "behaves like" a linear one, and this is translated by conjugation via a homeomorphism, although conjugation via a smooth diffeomorphism preserves some more subtle local invariants (resonances, e t c . ) . Coming back to control systems, first of all, the equivalent of conjugation by a smooth change of coordinates is (smooth) feedback equivalence, whose study was initiated in 4 , see a survey in 7 . In fact this is conjugation via a smooth diffeomorphism on the state and control, forced to have a triangular structure (see Proposition 2.5 below). The conditions under which a control system (1.1) is smoothly feedback equivalent to a linear controllable one are well known 8 ' 6 (and contrary to the case of ordinary differential equations, they are very simple), but they imply that very few nonlinear systems are locally feedback equivalent to a linear one, even when the linear approximation is controllable. This remark and the review of the situation for ordinary differential equations naturally brings about the question whether for control systems, relaxing the regularity of the conjugating maps, i.e. considering conjugacy by homeomorphisms instead of smooth diffeomorphisms would make more systems equivalent to a linear one. After recalling some basic facts in section 2, we give in section 3 an essentially negative answer to the question evoked above, based on quoting a result to appear in 3 , that topological conjugacy to a linear controllable system implies conjugacy by "almost" smooth feedback (but the gap is really small). Section 4 recalls, also from 3 , a technical open question that would allow a nicer result and a nicer description of that "almost" smooth conjugacy, and finally section 5 extends the discussion of the results from section 3, their implications, and the questions they raise in nonlinear modeling.

206

2 2.1

Preliminaries on equivalence of control systems Definitions

Consider two smooth control systems with state x (resp. z) and input u (resp. v):

x = f(x,u), x e i " , u e r , 1

z = g{z,v),

1

zeW '

, veW '

(2.1)

,

(2.2)

or, expanded in coordinates, Xi=fi(xi,...

,xn,ui,...

,um),

Zj=gj(zi,...

,zn-,vi,...

,vm<),

1 < i < n, 1 < j < n', with the /j's and g^s some smooth (i.e. C°°) maps. We assume that / and g are defined respectively on the whole of E n x Km and E" x E m because it simplifies many of the statements below; this is actually no loss of generality to us for all the results we prove are local with respect to x,u,z,v, so that / and g can be extended using partitions of unity outside some neighborhoods of the arguments under consideration without affecting the results. Definition 2.1. By a solution of (2.1) that remains in an open set ft C jjn+m ^ we mean a mapping j defined on a real interval : 7

: 7

~* ^

(2 3)

with 7i(t) € E" and 71 (t) £ Rm, such that 7 is measurable, locally bounded, 71 is absolutely continuous and, whenever [Ti,T2] C / , we have : 7i(T 2 ) - 71(^1)

=

/

2

f(li(t), Mt))

dt .

Solutions of (2.2) that remain in Q,' C M" + m are likewise defined to be mappings 7' : / —> ft' having the corresponding properties with respect to g. We now define the notion of conjugacy for control systems. Definition 2.2. Let X :

U

^n> (x,u) i-> x(x,u)

(24) = (xi(x,u),

XTL{X,U))

be a bijective mapping between two open subsets of ffira+m and K™ + m respectively. We say that \ conjugates 7 : 7 — ^ 0 and 7' : I —• ft' if and only if i = x ° 7-

207

We say that \ conjugates systems (2.1) and (2.2) if, for any real interval I, a map 7 : / —> SI is a solution of (2.1) that remains in SI if, and only if, X ° 7 is a solution of (2.2) that remains in SI'. We say that systems (2.1) and (2.2) are locally topologically conjugate at (0,0) if we can chose SI and SI' to be neighborhoods of the origin and x a homeomorphism. We say that they are locally smoothly conjugate if, in addition, x and X"1 are smooth. Here the word smooth means C°°. In case there is no control, so that m = m' = 0 and we omit u and xn, Definition 2.2 coincides with the classical notion of local topological conjugacy for non controlled differential equations, and may serve as a definition in this case too. Let us write more formally the classical local results on ordinary differential equations that we recalled in the introduction : Theorem 2.3 (Flow-box theorem). If m = 0 and /(0) ^ 0, system (2.1) is locally smoothly conjugate at 0 to the linear system z\ = 1, z^ = • • • = zn = 0. Theorem 2.4 (The Grobman-Hartman theorem). If m = 0, /(0) = 0, and /'(0) has no pure imaginary eigenvalue, system (2.1) is locally topologically conjugate at 0 to the linear system z = f'(0)z. Prom now on, we consider control system, i.e. we assume m > 1. 2.2

Some properties of conjugating maps

It turns out that conjugating homeomorphisms preserve the dimension of both the state and the control and must have a triangular structure : Proposition 2.5. With the notations of Definition 2.2, suppose that (2.1) and (2.2) are topologically conjugate via a homeomorphism x '• fi —> fJ'. Then n = n', m = m', and xi depends only on x: X{x,u)

=

( x i ( z ) , Xn(x,u))

.

(2.5)

Moreover, xi : ^R" —>• ^R™ is a homeomorphism. Proof. Let x, u, v! be such that (x, u) and (x, u') belong to fl. Let further x(t) be the solution to (2.1) with a;(0) = x and u(t) = u for t < 0 and u(t) = u' for t > 0. By conjugacy, z(t) = xi{x(t),u(t)) is a solution to (2.2) with v given by v(t) = xn(x(t),u(t)), for t 6 (—e,c) and some e > 0. In particular Xi(x(t),u(t)) is continuous in t so its values at 0 + and 0 - are equal. Hence Xi(x,u) = xi{x,u') so that xi : ^K" -> 0^„< is well defined and continuous. Similarly, ( x - 1 ) j induces a continuous inverse SI' , -» OR». • In view of Proposition 2.5, we will only consider conjugacy between systems having the same number of states and inputs. Hence the distinction between

208

(n,m) and (n',m') from now on disappears. Taking into account the triangular structure of \ m Proposition 2.5, one may describe conjugation as the result of changing coordinates in the statespace (by setting z = xi(x)) a n d feeding the system with a function both of the state and of a new control variable v (by setting u = (x - 1 )n(z,^)), in such a way that the correspondence (x, u) i-> (z,v) is invertible. In the language of control, this is known as a static feedback transformation, and two conjugate systems in the sense of Definition 2.2 would be termed equivalent under static feedback. This notion has received much attention, although only in the differentiable setting (i.e. when the triangular transformation ^ is a diffeomorphism), see e.g. 4 ' 7 . 2.3

Linearization

Recall that / is assumed to be smooth (of class C°°). Let us make a formal definition of topological and smooth linearizability. Definition 2.6. The system (2.1) is said to be locally topologically linearizable at (x, u) € R n + m if it is locally topologically conjugate, in the sense of Definition 2.2, to a linear controllable system z = Az + Bv. Definition 2.7. The system (2.1) is said to be locally smoothly linearizable at (x,u) € R" + m if it is locally smoothly conjugate, in the sense of Definition 2.2, to a linear controllable system z — Az + Bv. Explicit necessary and sufficient conditions for a nonlinear system to be locally smoothly linearizable at a point were given in 8 ' 6 , and also in 9 (the previous two references dealt with control affine systems only), and is recalled in many nonlinear control textbooks. Without mentioning these conditions, let us simply say that they require a certain number of distributions to be involutive, and that this is a very non-generic property. 3

Main result on topological linearization

Let us now give a —basically negative— answer to the natural question raised at the end of section 1 : for control systems, removing the differentiability requirement on the conjugacy does not allow many more control systems to be (topologically) conjugate to a linear controllable system, contrary to the situation of ordinary differential equations (without control), (see section 1 and Theorems 2.3 and 2.4). Recall that / is assumed to be smooth (of class C°°).

Theorem 3.1 ( 3 ). System (2.1) is locally topologically linearizable at (0,0) if, and only if there exists an open neighborhood ft of (0,0) in Rn+m and a

209 homeomorphism

x:

ft

-> fv

_

/ 3 Xx

(a;, w) (->• x(x,u) = ( x i ( z ) , Xn(z,")) (possibly different from the homeomorphism defining topological linearizability of the system) such that 1. x conjugates system (2.1) to a linear controllable system z = Az + Bv, in the sense of Definition 2.2, 2. Xi : H n —> Q'n defines a smooth (C°°) diffeomorphism. This does not state that topological linearizability implies smooth linearizability for x need not be a diffeomorphism even though xi is. In 3 , the conclusion of the theorem is called quasi smooth linearizability. A thorough discussion as well as the proof of Theorem 3.1 is given there. Let us recall here what is necessary to make this theorem clearer. Proposition 3.2. The conclusions of Theorem 3.1 imply that 1. BXTL • 0 -*• Km is smooth,

2. the rank of B is the maximum rank of df/du the origin.

in small neighborhoods of

Proof. Computing z at the origin of a trajectory starting from (x,u) 6 0 implies, by the smoothness of xi, ^-(x)f(x,u)

=

Axi(x) + Bxn(x,u) .

(3.2)

This gives an obviously smooth expression of Bxn- The second point is proved using Corollary 3.5° at points close to the origin where the rank of df/du is maximum, and hence locally constant. • If B is left invertible (i.e. has rank m), the first point implies that xn itself is smooth, and we have the following immediate corollary : Corollary 3.3. If there are points arbitrarily close to (0,0) where the rank of df/du is m (i.e. where this linear map is injective), then x in Theorem 3.1 is a smooth mapping. Of course if B has rank strictly less that m, xn need not be smooth. This is discussed in section 4. " T h e proof of Corollary 3.5 does not use Proposition 3.2.

210

Note that the assumption of Corollary 3.3 is very "reasonable": for instance for single input systems, the only case where it is not met is when / does not depend on u in a neighborhood of (0,0), but then the system cannot be topologically conjugate to a controllable linear system : Corollary 3.4. If m = 1, i.e. if (2.1) is a single input system, then \ »" Theorem 3.1 is a smooth mapping. This is however still not "smooth linearizability" because even though \ is smooth, its inverse might fail to be differentiable at the point of interest. The simplest example is the system x = u3, x e E , i i £ l

,

(3.3)

3

clearly conjugate by (z, v) = \(x, u) = (x, u ) to the linear controllable system z = v. Obviously, x ls smooth, x _ 1 is continuous, xi is the identity smooth diffeomorphism, but the inverse of x itself fails to be differentiable at the origin. In fact, no smooth diffeomorphism can conjugate these two systems. This can easily be proved but is also a consequence of the necessity part of the following result that tells us exactly when smooth linearizability is implied by topological linearizability : Corollary 3.5. When f is of class C°°, system (2.1) is locally smoothly linearizable at (0,0) if and only if it is locally topologically linearizable at (0,0) and the rank of df/du is constant around (0,0). Proof. Smooth linearizability is a particular case of topological linearizability, and it implies constant rank of df/du because differentiability of the smooth diffeomorphism and its inverse allow one to get a formula for df/du(x,u). Let us prove the converse. Suppose that the rank of df/du is r < m in a neighborhood of (0,0) and that system (2.1) is locally topologically linearizable at (0,0). From Theorem 3.1, this implies that there exists a triangular homeomorphism (x,u) >-)• (z,v) = x(x,u) = (xi(x),xn(x,u)) that conjugates system (2.1) to a linear controllable system z = Az + Bv with the additional property that xi defines a smooth diffeomorphism from a neighborhood of 0 € M™ onto its image. Let r' < m be the rank of the matrix B. There are invertible n x n and m x m matrices P and Q such that ' Ir, Bc

= PBQ

=

(3.4) 0

V where Iri is the r' x r' identity matrix.

o\ 0

211

Computing z at the origin of a trajectory starting from (x, u) £ Q implies (3.2) by the smoothness of xi- Hence the map BcQ~xXi is smooth where it is denned, and differentiating (3.2) with respect to u yields :

Since P^-(x) is invertible and the rank of df/du is r, both sides have constant rank r. This implies r < r'. This also implies that the mapping Rn+m _j. K n+ r ' defined by (x,u)

H->

(x,

BCQ~1XTL{X,U))

,

has a constant rank n + r i n a neighborhood of the origin, hence, by the constant rank theorem applied to this mapping, there i s a r x m matrix K of rank r (selects r lines that are independent among the m lines of (x,u)), 9u n+m r a neighborhood ft of (0,0) in R , two smooth mappings a : K"+ -> Rn+m and ft : Rn+m -> Mr ~ r (in fact they only need to be defined in suitable neighborhoods of the origin) such that x is defined on fi, and BcQ~lU-,u)

=

(«(*,Jrfleg-ix.(z,«))J

(3.5)

for all (a;, u) £ fi and (x,u)

*-> ( a r . J f B c Q ^ x i t a r . u J . ^ i . u ) ) ,

(3.6)

defines a smooth diffeomorphism from ft onto its image. This implies that r = r' because from (3.5, r < r' would prevent x from being one-to-one. Hence K can be taken the identity matrix. Define x : fi -* IR"+m by \ — L ° V' with ^(x,w)

(Fxi(a;), i f 5 c Q _ 1 x n ( 3 ; , u ) ,

=

1

0(x,u))

1

and L(z,v) = (P~ z,Q~ v). ijj is a smooth diffeomorphism because (3.6) is one, and L is obviously a (linear) smooth diffeomorphism. Setting (z, v) — x(x, u) conjugates system (2.1) to z = Az + Bv. D 4

A n open question

It is a reasonable question to ask whether the conclusion of Corollary 3.3 holds in general, namely whether Theorem 3.1 can be strengthened so as to state that x is, on top of its other properties, a smooth mapping (when the rank of df/du is not locally constant, x would fail to be be differentiable, from the necessity part of Corollary 3.5).

212

Let us examine the case where the assumptions of Corollaries 3.3, 3.4 and 3.5 fail (these three corollaries already state the desired conclusion), namely the case of systems with m < 2 controls where the rank of df/du is everywhere strictly smaller than m, studied locally around a point where this rank is not constant (i.e. the rank at the point is strictly less than the maximum rank in arbitrary small neighborhoods of this point, itself strictly smaller than m. The smallest dimensions where this occurs is n = 1, m = 2, i.e. systems x = f(x,ui,U2) with x, ui and u2 scalar. In order to state our open question in the smallest dimension possible, let us drop the dependence on the righthand side on x and consider systems x = a(ui,u2),

i £ l , u = (tti,u 2 ) £ E 2 ,

(4.1)

2

where a : E -» E is smooth. Let us assume that this system is locally topologically linearizable around (x,u) — (0,0,0). The only canonical controllable linear system with one state z € IK and two controls (v\,v2) 6 E 2 is z = vi, hence local topological linearizability means existence of a homeomorphism X : (x,u1,u2)

>-• (z,v1,v2)

= (Xi(x),

\2(x,ui,u2), an

ls

X3(x,ui,u2))

(4.2)

tnat

(in the terms of Definition 2.6, xi is xi d Xi (X2,X3)) conjugates (4.1) to the linear system z — v\. From Theorem 3.1, this implies existence of another homeomorphism x of the same triangular form, that we denote by X instead of x, such that xi is a smooth diffeomorphism (from a real interval containing zero onto an open interval) and X2 is a smooth mapping from an open neighborhood of the origin in E 3 to E, while our results do not grant that X3 has any more regularity than continuity. In fact the conjugation reads dxi -jr— (x)a(ui,u2) - X2(x,ui,u2). (4.3) This implies in particular that X2 does not depend on x, and then one can replace v2 = X3(^7^1,1*2) with v2 = X 3 ( 0 J M I > U 2 ) without changing the conjugating property. Composing x given by (4.2) with (z,Vi,v2) H* (XI _ 1 iz), i l r (Xi_1 (•z)) _1 ^i. v2), one finally gets a conjugating homeomorphism of the form (x,u1:u2)

H-» (x, a(ui,u2),

P(ui,u2))

(4.4)

where f3(u\,u2) = X3(0,ui,U2)- Hence local topological linearizability amounts to existence of a continuous mapping /? from an open neighborhood of the origin in E 2 to E such that (u\,u2) i-> (a(ui,u2),/3(ui,u2)) defines a homeomorphism from a neighborhood of the origin in E 2 onto its image (we just proved it is necessary, but conversely, it makes (4.4) a local homeomorphism, that obviously conjugates (4.1) to z = vi). Similarly, conjugacy via

213

a homeomorphism that is a smooth map amounts to existence of a smooth mapping having the same property. Hence the question whether \ c a n D e taken a smooth mapping in Theorem 3.1 reduces to the following Open Question 4.1. Let a and /3 be two mappings ] — e,e[2 —> E, e > 0, such that a is smooth, j3 is continuous, and (111,112) *-> {a(ui,U2),fi(ui,U2)) defines a homeomorphism from ] — e,e[2 onto its image. Does there exist a smooth mapping b :] - £',e'[2-t R, 0 < e' < e, such that (1^,1*2) >-> (a(ui,U2),b(ui,ii2)) defines a homeomorphism from]—e' ,e'[2 onto its image ? This question in differential topology can be posed in higher dimension of course, see below. It is of interest in its own right and seems to have no answer so far, even for p — q = 1. Open Question 4.2. Let O be a neighborhood of the origin in W+q and F : O —»• W a smooth map. Suppose there is a continuous map G : O —• K? such that F x G : O —>• Rp x l ' is a local homeomorphism at 0. Does there exist another neighborhood of the origin O' C O and a smooth mapping H : O' ->• W such that F x H : O' ->• E p x W is again a local homeomorphism atO? 5

Implications in Control Theory

Let us come back to the discussion we started in the Introduction. Consider a control system (1.1), assume for simplicity that we work around an equilibrium, i.e. /(0,0) = 0, and let us write its linear approximation, i.e. f(x,u)

= Ax + Bu + F(x,u) dF dF with F(0,0) = — (0,0) = — (0,0) = 0 ,

(5.1) (5.2)

so that the nonlinear system (1.1) reads x

= Ax + Bu + F(x,u)

.

(5.3)

From the remarks made in the introduction, the relevant situation is the one where the linear system i

=

Az + Bv

(5.4)

is controllable. Let us assume slightly more to stay away from the pathologies evoked in the previous section. The additional assumption is very mild and is, for instance, always true when the constant n x m matrix B has rank m; it is implied by linear controllability for single input systems. Assumption 1. The pair {A,B) is controllable and the rank of B + dF — (x,u) is equal to the rank of B for small (x,u).

214

The question raised in the introduction was the one of finding a reasonable equivalence relation that would make the two systems (5.3) and (5.4) locally equivalent. Comparing the situation of ordinary differential equations (without control), a candidate was local topological conjugacy as in Definitions 2.2 and 2.6, and if that candidate was successful, we would have a result making precise the vague statement (1.2). Corollary 3.5 implies that, for A, B and F satisfying Assumption 1, systems (5.3) and (5.4) are locally topologically conjugate if and only if they are locally smoothly conjugate, and it is known from 8-6 that this is false for a generic F, even satisfying (5.2). This discards topological conjugacy as a candidate for the above mentioned equivalence relation, but this does not contradict the basic belief behind statement (1.2). A way to contradict that statement would be to find at least one example satisfying the assumption, but where the nonlinear system (5.3) displays some local "qualitative" phenomenon that do not occur for the linear system (5.4). In the qualitative theory of dynamical systems (without control), the phase portrait gives a picture of the behavior, on which phenomena like attractors, invariant set, (stable) closed orbits can just be "seen". A control system is more complex : it describes how the behavior of the state (at least in the state space representation) is linked to the control. It is not very clear what a qualitative phenomenon should be for a control system. The least to require is that it be invariant by topological conjugacy as defined here. In the introduction, we pointed out that (non-)controllability is a qualitative property, but it is of no help here since (1.2) only refers to controllable systems. We do believe that clarifying the status of a statement like (1.2) is very relevant to control theory and modeling. Our negative results (section 3) say that topological conjugacy is not the right tool to answer this. A looser equivalence could be a way to state (1.2) properly. It could also be that the intuition behind (1.2) is totally wrong and that some nonlinearities F allow system (5.3) to display some qualitative phenomena locally that cannot occur on a linear system (5.4).

Acknowledgements The authors wish to thank Prof. I. Kupka, from Universite Pierre et Marie Curie (Paris VI) for many enriching discussions. Thanks are also due to Prof. C.T.C. Wall from the University of Liverpool for his comments on the "Open Question 4.2" in section 4.

215

References 1. V. I. Arnold, Equations Differentielles Ordinaires, 3 ed. (MIR, 1974). 2. V. I. Arnold, Chapitres supplementaires de la theorie des equations differentielles ordinaires (MIR, 1980). 3. L. Baratchart, M. Chyba, and J.-B. Pomet, On the Grobman-Hartman theorem for control systems, in preparation. 4. R. W. Brockett, Feedback invariants for nonlinear systems, IFAC World Congress (Helsinki), 1115-1120 (1978). 5. P. Hartman, Ordinary differential equations, 2 ed., (Birkhauser, 1982). 6. L. R. Hunt, R. Su, and G. Meyer, Design for multi-input nonlinear systems, in Differential Geometric Control Theory, R.W. Brockett, Ed., 258298, (Birkhauser, 1983). 7. B. Jakubczyk, Equivalence and invariants of nonlinear control systems, in Nonlinear Controllability and Optimal Control, Hector J. Sussmann, Ed., (Marcel Dekker, New-York, 1990). 8. B. Jakubczyk and W. Respondek, On linearization of control systems, Bull. Acad. Polonaise Sci. Ser. Set. Math., 28, 517-522 (1980). 9. A. J. van der Schaft, Linearization and input-output decoupling for general nonlinear systems., Syst. & Control Lett., 5, 27-33 (1984).

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217

LOCAL A P P R O X I M A T I O N OF T H E R E A C H A B L E SET OF CONTROL PROCESSES ROSA MARIA BIANCHINI Department of Mathematics U. Dini, University of Florence, Viale Morgagni 67/A, 50134 Firenze, Italy E-mail: bianchiniQmath. unifi. it This paper concerns the variational approach to the study of controls problems. We construct local approximations of the reachable set from which sufficient conditions for a trajectory to be locally controllable and necessary conditions for a reference pair to be a weak solution of a Mayer control problem with end constraints can be derived.

1

Introduction

The knowledge of the geometry of the reachable set can help greatly in several control problems. In particular what is really useful is to find conditions which the boundary points have to fulfill. For example a reference trajectory is locally controllable if its final point is not a boundary point, which surely happens if the point does not satisfies the necessary conditions. An other example is given by Mayer optimization problems without constraints on the final point: under standard hypothesis, if a reference trajectory solves the optimization problem then its final point is a boundary point of the reachable set and hence has to satisfies the necessary conditions. This is the basis of Pontriagin Maximum Principle and of its generalizations 1, 2 etc. Since it is difficult to compute directly the reachable set, one tries to approximate it near a point x by means of tangent vectors, objects which can be computed easier. But the collection of tangent vectors provides a good approximation of the set only in the case in which the boundary is sufficiently regular; for an example it works if the set is locally convex. Unfortunately the boundary of the reachable set may be irregular even in the analytic case, see 1. This has been the starting point for searching subsets of tangent vectors which could provide the informations one needs in concrete problems, see 3 , 1 and the references therein. In this paper I will start with reviewing some local approximations of the reachable set of a control process pointing out their relations with the study of some control problems. Later I will construct a local approximation of the reachable set of an affine control process near the final point of the trajectory

218

relative to the drift term; this local approximation can be used to test either the local controllability of the trajectory or its optimality respect to a Mayer control problem with constraints on the end points. In the last section I will show by means of some examples how the theory can be applied. The natural setting for this variational approach, is the one of differential geometry; the control process are family of differential equation on a differential manifold, M, finite dimensional, sufficiently regular and the tangent vectors at y is an element of the tangent space of M at y. However for simplicity sake I am going to examine the particular case in which the manifold is an Euclidean space. 2

Tangent cones

Let us consider a control process x = f(t,x,u)

u(-)EU

(2.1)

where the state x belongs to M" , the function / and the set of admissible control U are such that the following standard assumption is verified A) for each admissible control u(-) e U the Cauchy problem (x =

f(t,x,u(t))

\ X(t0) = X0

has an unique maximal solution whatever is the initial condition. We will denote by S(t,to,x0,u) the value at time t of the solution of (2.1) relative to the control u which at time to takes the value xo and by R(t, to, xo) the reachable set at time t from the initial conditions (to,£o) R(t,t0,x0)

= | S(t,t0,x0,u)

:

ueU>.

We are interested in the following problems: LC Local controllability of a trajectory, x* : t -» [£o>£i], i-e. x*(ti) € m«i?(ti,t 0 ,'a;*(
Vt > 0.

Mayer optimization problem, i.e. locally minimize the value of a C 1 functional, /o, over the set R(ti,t0,x0).

219 COP

Mayer optimization problem with constraints on the final point, i.e. locally minimize the value of a C 1 functional, /o, over the intersection of the set R(ti,t0,x0) with a given subset C.

A possible approach to these problems is to derive either necessary or sufficient conditions from a local approximation of the reachable set at the final point of the trajectory 1 4 5 6 7 . The local approximations used in the quoted papers are suitable subsets of the intermediate tangent cone, a subset of the contingent tangent cone; the difference among these papers depends on the subsets considered. I want to make some comments about this approach and I want to state the results I am going to use in the next paragraph. Let us start with the definition of the intermediate tangent cone to a subset of an eucleadian vector space, 8 : Definition 2.1. Let K be a subset of Mn and let y E K. The itermediate, or adjacent, tangent cone to K at y, IK{V) , is the set of vectors tangent at y to the curves contained in K passing from y IK(V) = [v

: y + ev + o(e) £K

ee[0,e]|.

The vectors belonging to IK{V) are named tangent vectors to K at y. For studying the LC property, the intermediate tangent cone to the reachable set at the final point of the trajectory is a too large set because it can be the whole space even if the property does not hold, see the example in x. However one can derive sufficient conditions for this property to hold by considering only subsets, H, of lR(tltt0,x0)(x*(Ti)) chosen in such a way that the following property holds: A) if the set H contains a positive basis of tangent vectors, then the point is an interior point of the reachable set. I recall that a set of vectors is positive basis of W1 if 0 belongs to the interior of convex hull of the set. Notice that condition A) could be replaced by the weaker condition: A ' ) if the set H is the whole space, then the point is an interior point of the reachable set. however property A) is more useful in the applications. Let us consider the optimal problems. The first thing to notice concerning these problems, is that the set /fi(t1,t0,a:o)(2/) is constructed by means of the points close to y which can be reached from (t0,x0) at time t\ independetly from the trajectories by means of which this can be done, meanwhile in the

220

definition of the O P and of C O P problems, locally has to be understood in the sense of the calculus of variations, i.e. either strong or weak solutions of the problem. Let me state the definitions of strong and of weak solutions which I am going to use. An admissible pair of a control process in a given interval [to,*i], is a couple (x* ,u*), x* solution relative to u* on [to>*i] • Definition 2.2. A pair (x*,u*) is a strong solutions of O P if the final point of the reference trajectory, x*(ti), is a point of minimum of the functional /o on the set of points reachable from the initial point by means of a trajectory which are near the trajectory x* in the C°([£o;*i]] topology. Weak solution of O P may be defined if it is given a topology on the set of control with respect to which the solution map is continuous. In the sequel we will assume that such a topology, II, is a data of the problem. Definition 2.3. A reference pair (x*,u*) is a weak solutions of O P if the final point of the reference trajectory, x*(ti), is a point of minimum of the functional /o on the set of points reachable from the initial point by means of pairs (x, u) belonging to a neighborhood of the reference pair in the C°([to, ti]) x U topology. For taking in account this fact, only particular tangent vectors to R(ti,to, XQ>) at x*(ti) have been considered; roughly speaking we look for the tangent vectors to the curves obtained by slightly modifying the control u*. Tangent vectors of this type are the ones introduced by A. Bressan 5 for the affine systems. I will name these tangent vectors, weak trajectory variations because they depend on the reference pair considered, not only on the final point of the trajectory. The definition of the weak trajectory variations for affine system is the following Definition 2.4. Let t \-t x*{t) be a solution of the C1 control process m

X = fo(x) + Y2Uifi(X) on the interval [to,ti] relative to the control u*. A vector v is a weak trajectory variation of (x*,y*) if there exist a family of admissible control maps, u(e) , e G [0,e] continuous in the L1 topology, such that 1. l i m ^ 0 + / £ \\u{e){s) - u*(s)\\ ds = 0 2. 5(*i,i0,a;*(*o),M(e)) = x*{tx) + ev + o(e). Notice that since the solution map of an affine control process depends continuously from the control in the L1 topology, from property 1 it follows that lime_+0+ \\S(-,t0,x*(t0),u(s)) -x*(-)))\\ = 0 in the C° norm, and hence

221

the weak trajectory variations are tangent vectors at x*(*i) to the set of points which can be reached from (to,xo) by means of trajectories which lay near the reference one. The definition of weak trajectory variations can be generalized to any control process if there exists a topology on the set of controls with respect to which the solution map restricted to a compact interval is continuous. If (x*,u*) is a weak solution of the O P problem, then Dfo{x*{h))v

>0

Vu weak variation of {x*, u*).

(2.2)

This implies that if we find a positive basis whose elements are weak trajectory variations of (x*,u*), then (x*,u*) is not a weak solution of O P . An example provides by A. Bressan in 5 proves that if there are constraints on the final point, then condition (2.2) is no longer a necessary condition for (x*,u*) to be a weak minimum. In fact in that example the reference pair is given by x*(t) = 0, t € [0,3], u* = 0; the reachable set from the origin at time 3 is i?(3,0,0) = { ( x , 2 / , 2 ) e K 3

:

xy>0}

the set of weak trajectory variations of (x*,u*) contains a positive basis, but if C = {x — —y, z = 0 } then (x*,u*) is a weak minimum of C O P independetly from the cost / 0 . The condition which a weak solution of C O P has to fulfill is £>/o(x*(ti))w>0

(2-3)

for each weak trajectory variation w , which belongs to the intermediate tangent cone of the intersection of the reachable set with the set of constraints, C. The intermediate tangent cone to the intersection of two sets does not coincide with the intersection of the tangent cones of the sets considered. This implies that (2.3) holds only for the vectors belonging to a collection of trajectory variations, H, which have the following property 1,3: B) if the convex closure of the set H cannot be separated from a convex cone V, and if coH and V are not two complementary subspaces, then the set V n co H contains at least one half line. The previous arguments lead to search for subsets of weak trajectory variations which have either property A) or property B ) . Bressan, in the quoted paper, has single out subsets of weak trajectory variations which have both the properties. A point to underline about this result, is that these subsets are particular collections of weak trajectory variations and they are not a collection of particular weak trajectory variations.

222

Other authors have chosen a different way to select weak trajectory variations with the required properties. They introduce a more restrictive definition of trajectory variations, 6 , 7 , 9 ecc, the needle-like variations. We will assume that it is given a topology on the set of controls with respect to which the solution map is continuous and continuously differentiable with respect to the initial state. This hypothesis is not necessary for introducing the needle-like variations, but it let the definition simpler. For conditions which insure that this hypothesis is fulfilled see 10 . Definition 2.5. Let (x*,u*) be a reference pair on the interval [to,ti] and let T € [io^i] • A vector v is a needle-like trajectory variation of x* at r if for each e £ [0,e] there exists an admissible control map u(e) with the following properties: 1. e >-> u(e) is continuous 2. u(e) coincides with the reference control outside the interval, [a(e),b(e)\ 3. lime_^0+ He) = lime_>0+ a(e) = r 4. S(ti,to,x*{to)),u(e))=x*(ti)

+ ev + o(e).

More general needle-like variations can be denned. I have chosen this definition because it is rather simple but sufficiently general to illustrate the main properties of this type of variations. The needle-like variations at r indicates the controllable directions of the reference trajectory from X*(T) . Notice that, if the values of the control u(e) are uniformly bounded, then from 3 it follows that u(e) tends to u* in any Lv norm, therefore for a large class of control processes 10 the needle-like variations are weak trajectory variations. It is an open question if the set of needle-like variations has property A)or property B)but both the properties has been proved for some subsets of them. It is known that any collection of needle-like variations each occurring at a time different from all the others has the properties we are speaking about. This implies that also any set of the needle-like variations which are limit point of sequence of variations each occurring at a different time, have the properties A)and B ) , see 6 , 7 . We will call this type of variations, K-variations. More precisely the definition of K-variation at time r is the following: Definition 2.6. A vector v is a K-variation atr if there exists a non stationary sequence {r„} converging to r and for each r n a needle-like variation, vn at rn, such that {vn} converges to v. The K-variations depends on the behavior of the control process along an arc of trajectory. In 4 G. Stefani and myself have single out a subsets of needle-

223

like trajectory variations which are of local nature and which have property A and property B. In the next section I am going to use these variations in a simplified version which can be found in u . Let us start with the definition of good variations. Definition 2.7. A vector v is a right good-variation, (left good-variation) of the pair (x*,u*) at r if there exists positive numbers k,c,e and for each e € [0, e] a family of admissible control maps, ue (c), c € [0, c] with the following properties: 1. we(c) coincides with the reference control outside the interval [r + e ,T + (1 + a)ek] 2. for each e, c H-> ue(c) is continuous 3. S(T+(a+l)ek,

r+ek,

(S(r-ek, r-{a+l)ek, uniformly w.r.t. c.

x*(r+ek),

ue(c)) = x*(T + {a+l)ek) + cev + o(e)

x*(T-(a+l)ek),

us(c)) = x*{T~ek)+cev+o(t)

)

The good variations will be simpler named g-variations. Let G(t, T) denote the transport along the flow generated by the reference control from x*(r) to x*(t); G(t,r) is the evolution matrix of the linearization along the reference trajectory. If v is a g-variation at r, then G(ti,r)v is a needle-like variation at r. Definition 2.8. The varational cone of the reference pair (x*,u*), /C, is given by /C = co< G(ti,r)v

,

v g-variation at

T\

It n it is proved that K, has both the properties A)and B)and that, under mild hypothesis, it contains the set of K-variations. Properties A)and B)imply the following theorems: Theorem 2.9. Let x* be a trajectory relative to the control u* defined in the interval [^o^i] • If the set of g-variations of (x*,u*) at to contains a positive basis of the state space, then the trajectory is STLC. / / the variational cone coincides with all the tangent space at x*{t{), then the trajectory x* is LC in the interval [to,ti]. I want to remark that it can be proved that if the variational cone is the whole space, then the trajectory x* is LC also in the interval [£o, *i — e], for e sufficiently small. This last property imply, for a large class of control process, that if x* is time optimal in this interval, then its variational cone cannot be the whole tangent space.

224

Theorem 2.10. Let {x*,u*) be a weak solution of COP and let N be the Dubovitskij-Miljutin tangent conea 8 of the constraints set C at x* (t\). Under these hypothesis there exists Ao < 0 and a linear functional v at least one of which different from zero, such that vw>0, (XoDfoix*^))

s

+ V) y < 0

Vw<EN

(2.4)

Vy e K

(2.5)

Prom the previous theorem one can derive high order Maximum Principle. Let me give an example of this type of results. I recall that if the control process is C 1 and if the values of the controls are bounded, then the "right" topology for the set of controls is the Lp([t0,ti]) one. This implies that the Pontriagin variations are g-variations. From Theorem 2.10 it follows: Theorem 2.11. Let

x = f(t,x,u)

u € {/

be a C1 control process and let U be a bounded set. If (x*,u*) is a weak solution of COP and if Dfo(x*(ti)) ^ 0, then there exist a non trivial solution X(t) of the adjoint equation

X=

-Xfx(t,x*(t),u*(t))

such that X(ti) = AoD fo(x* (ti)) + v X(t) v < 0

Ao

and

v

as in Theorem 2.10 W, g-variationat T

and

X(t)f(t,x*(t),u*(t))

=maxX(t)f(t,x*(t),u)

a.e. t£

[t0,ti].

Notice that a strong solution of a COP is also a weak one therefore the previous theorem is also a necessary condition for the pair be a strong solution. a

t h e vector v belongs to this tangent cone if for all s sufficiently small I * ( ( I ) + £ » G C

225

3

Examples of g-variations

In this section I am going to consider a single input, C°°, affine control process (£)

x = f{x) +ug(x)

|u| < S

(3.1)

the admissible controls are the locally integrable maps satisfying the constraints ||u(i)|| < S endowed with the L\oc topology. In the following the reference pair will be (x*,0) with x* : [0,T] -> W1 , t !->• exp t / o O , i.e. the reference trajectory is the one relative to the drift term of the process and the reference control is the null one. g-variations of this pair have been constructed by means of the "neutralization of the obstructions". To recall this result I need to introduce some notations. Let LieX denote the free algebra generated by two indeterminate .Xo, X\; if Z,Y £ LieX, [Z,Y] denotes the Lie bracket of Z and Y and adkzY is defined inductively by ad°zY = Y,

ad%+1Y = [Z, adzY].

Let A € UeX; |A|j is the number of times that Xi appears in A. Definition 3.1. Sussmann 12, Bianchini-Stefani 13 The bad brackets are the brackets in Lie X which contain XQ an odd number of times and X\ an even number of times. Let B be the set of bad brackets B = {A , |Ajo is odd |A|i is even }. The set of the obstructions is the set B* = Lie (X0, B) \ {aX0 , a G R}. The name of obstructions come from the fact that if all these brackets are equal to zero at the initial point of the reference trajectory, then the trajectory is STLC and hence the final point of the reference trajectory is an interior point of the reachable set. The same result holds if the obstructions which are different from 0, are neutralized by means of a "weight". Definition 3.2. An admissible weiaht for the process (3.1) is a couple of non negative numbers, I— (lo, l\), which verify the relations 0 < lo < l\ . Definition 3.3. Let 1 = {lo,h} be an admissible weight, the \-weight of a bracket $ is given by ||*||i = 'o|*|o + *i|*|iAn element 0 £ Lie X is said l-homogeneous if it is linear combination of brackets with the same \-weight, which we name the l-weight of the element.

226

The weight of a bracket, $ , with respect to the standard weight 1 = {1,1} coincides with its length and it is denoted by | | $ | | . Let 0 6 Lie X; with 0 s we denote the element in the Lie algebra generated by the vector fields / , g obtained from 0 substituting to XQ the vector field / and to Xi the vector field g. Definition 3.4. Let 0 € Lie X; following Sussmann 12 we say that 0 is ly.-neutralized at a point y if the value at y of 0 s is linear combination of the values of brackets with less l-weight, i.e. &T,(y) — 5Z ai ^s(2/) > 11 ^ J 111 <

lieili. The result in 4 states that the directions individuate by evaluating at r those elements of the Lie algebra associated to the system with the property that each obstruction which has less or equal l-weight is ls-neutralized at r , is a g-variation at r. Theorem 3.5. Letr € [0, T] and let A € LieX ; if there exists an admissible weight 1 such that each obstruction whose l-weight is less than or equal to \\A\\\ is ly.-neutralized at X*(T) then ±As(a;*(r)) are g-variations at r . This theorem implies that to each admissible weight 1 and to each time r we can associate a subspace of g-variations at r, the subspace spanned by AY,(X*(T)), A 1-homogeneous element whose weight is not greater than a positive number, p(l, r ) . In 14 it is proved that if the neutralization of the obstructions holds on an arc of trajectory containing x* ( r ) , then the set of g-variations which can be associated to 1 and r, is a larger set since to the subspace described above one can add an other g-variation whose direction is given by obstructions; this further variation, v, is of unilateral type, i.e. —v may be not a g-variation. Notice that variations of unilateral type are very important in optimization theory since they distinguish between minimization and maximization problems. I refer to 14 for the statement of the general result because it is very technical; I limit myself to quote two results which can be deduced from it. Further investigation are necessary to derive from the general result other explicit computable g-variations. Proposition 3.6. Let 1 be an admissible weight such that each obstruction whose l-weight is less than (2kli + lo) is Is -neutralized on an arc of trajectory containing x* ( r ) , then

(adlkf)(x*(t)) is a g-variation at T . Let me remark that since whatever is 1 there are no obstruction of weight

227

less than (2Zi + lo), then (ad2g)f(x*(t)) is a g-variation at each t £ [0, T]. Proposition 3.7. If there exists an admissible weight 1 such that each bad bracket of l-weight less than (2n + l)Zo + 2Zi is l-neutralized on an arc containing x* (T) , then (-ir+i[g,adf+'g](x*(T)) is a g-variation at T. The Legendre-Clebsch condition can be deduced from the previous proposition. In fact the neutralization by means of the weight 1 = (0,1) is given by brackets which contain a less number of g so that the following proposition holds. Proposition 3.8. Let Li(x*{t)) = span {Ax(x*(t)) : ||A||i = 1} and let p be an integer with the property that for each t in a non degenerate interval, J, contained in [0, T] ; [5,[a4,5]](x*(i))€Li(x*(t))

:t = l,...,p-l

for each t E J. If \g, [adpp] {x*(t)) $ Li(x*(t)), and

(3.2)

then p is an odd integer

(-l)ip+1)/2[g,[^f,g)]{x*(t)) is a g-variation at t for each t £ J. Remark 3.9. It can be proved u that the neutralization on an arc of trajectory implies that the integer p has the property that if A 6 Lie X contains two Xi and i times Xo, i < p, then A, is Is -neutralized on the interval. In Propositions 3.6 and 3.7 it is required the neutralizations of some obstructions on an arc of the reference trajectory, meanwhile in Theorem 3.5 only the neutralization in one point of the trajectory is required. I don't know if the neutralization in only one point by means of any admissible weight is sufficient for constructing unilateral g-variations, but in a forthcoming paper I am going to prove that, if the neutralization is done by means of the weight (0,1), then to the subspace of g-variations given by Theorem 3.5, one can add an unilateral g-variation. The analogous of Proposition 3.6 is Theorem 3.10. If r 6 [0,T] and each obstructions which contains (2p— 2) times g is neutralized at x* (T) by means of brackets which either contain a less number of g or contain an equal number of g but a less number of f, then

(adff){x*{T))

228

is a g-variation at r . Up to now it is not clear what is the analogous of Proposition 3.7. 4

Applications

In this section I want to show how the theory developed in the previous sections can be applied. Let me start with a L C problem. We want to know if there exists T > 0 such that the stationary trajectory x* (t) = 0 of the system ' i\ = u X2=X! X3 = X4

^ X4 = -X3 + x\ \u\ < 1, is locally controllable in the interval [0, T]. Let us calculate the variational cone of the pair (x*, 0) in the interval [0, T]. Since the trajectory is a stationary one any neutralization holds on the whole trajectory and the only obstructions which have to be neutralized are the bad brackets, 13 . The only bad brachets non vanishing at 0 contain two g and three / ; they cannot be neutralized by any weight, g = d/dxi and [f>d] = —d/dx2 have less 1-weight of these bracketts whatever is the weight 1, therefore the subspace span{ei, e 2 } is a subspace of g-variations at each time; [/,g], [[f,g],f]\x*(t) ^ 0 implies that the number p of Proposition 3.7 is equal to 3; therefore v = 9J [/>[/>[/>d]]] \x*(t) = 2d/8x4 is an unilateral g-variation at each r. It can be proved that the trajectory cannot be STLC; however if T > TT then the conic convex hull of the set

U

GT T

(>>

T€[0,T]

is the subspace spanned by {e 3 , e^] . Therefore if T > ix, then the variational cone is equal to M4 . We can conclude that the trajectory is locally controllable in any interval larger than IT. Let me notice that we cannot a priori exclude that the trajectory is locally controllable in a shorter interval of time, but it is really probably that it is not.

229

Let us examine now an application to an optimization problem with constraints on the end point. Consider the following C O P : minimize the functional fo(xi,X2,x3,X4,x5,X6) = -x5 + x\ over the set of points reachable in time 1 by means of the trajectories of the control process ±1 = 1 X2 = U X3 = X4

xh XQ

X2

Xz Xo

Xn

X^ -p X3

—

starting at 0 and satisfying the constraint x5+x6

(4.1)

= 0.

Let us calculate the variational cone of the pair (x*(t) = 0, u*(t) = 0) in the interval [0,1]. The obstructions which do not vanishing along the trajectory contain two g and at least three / , therefore span < e%, e3, a \ , the subspace spanned by brackets which contains only one g and at most 2 / , is a subspace of g-variations at any t € [0,1]. The obstruction adj. -,f(x*(t)) is proportional to the bracket adzgf{x*{t)); therefore each obstructios which contain two g and at most four / is 1-neutralized along the trajectory by means of the weight (1,1), 14 . By Proposition 3.7, ±ad3gf{x*(t)) = ±6d/dx5, -[g,[ad5fg\] = —2d/dxe are variations at any t. The linearization of the process along the reference pair is given by: 2/1=0 2/2 = 0 2/3 = 2 / 2 2/4 = 2 / 3

2/5 = 0 2/6=0 therefore the variational cone of the reference pair is the half-space: { x$ < 0} . If the reference pair is a weak solution of C O P , then by Proposition 3.7 there exist y € C1- and A < 0 such that (AV/ 0 (a;*(l)) + 2/, v) < 0 T

(4.2)

v G /C. T

Since Vf0(x*(h)) = (0,0,0,0, - 1 , 0 ) and y = ( 0 , 0 , 0 , 0 , a , a ) , for (4.2) to hold there exists a number a > 0 and a number A < 0 such that the vector (0, 0, 0, 0, - A + a, a)T

230

is orthogonal to span {e 2 , e%, e\, e 5 }. Since this is impossible, the reference pair is not a weak solution of the problem. 5

Open problems

The main open problem of this theory is the following: is it necessary to limit ourselves to consider only very particular tangent vectors to the reachable set for the variational approach either of COP or of LC problem? Taking in account what is known, I suggest to investigate to these specific points: 1. The weak trajectory variations has not the property A)and hence cannot have the property B)). Does any its convex subset has property B)? Is it true that if the set of weak trajectory variations is the whole tangent space, then the trajectory is LC? 2. Is the set of needle-like variations at a given time a convex set? 3. Have any convex subset of needle-like variations property A) and/or property B)? References 1. Bianchini, R.M. , Variational Approach to some Optimization Control Problems, in Geometry in Nonlinear Control and Differential Inclusions, Banach Center Publication, Warszawa, 32, 83- 94 (1995). 2. Mirica, S., New proof and some generalizations of the minimum principle in optimal control J. Optim.Theory Appl., 74, 487-508 (1992). 3. Mirica, S., Intersection Properties of Tangent Cones and Generalized Multipler rules, Optimization, 46, 135-163 (1999). 4. Bianchini, R.M., and Stefani, G., Controllability along a Reference Trajectory: a Variational Approach, SIAM Journal on Control and Optimization, 3 1 , 900-927 (1993). 5. Bressan, A., A high order test for optimality of bang-bang controls, SIAM Journal on Control and Optimization, 23, 38-48 (1985). 6. Krener, A. , The High Order Maximal Principle and its Applications to Singular Extremal, SIAM Journal on Control and Optimization, 15, 256-292 (1977). 7. Knobloch, H.W. , Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Control and Informations Sciences, 34, (Springer-Verlag, Berlin, 1981).

231

8. Aubin, J.P. , Frankowska, H., Set-valued Analysis (Birkhauser, Boston, Base, Berlin, 1990). 9. Agrachev, A.A. , Newton Diagrams and Tangent Cones to Attainable Sets, in Colloque international sur I'analyse des systemes dynamiques controles, 3-6 Juillet, Lyon, France, 1-12 (1990). 10. Bianchini, R.M. , and Margheri, A., First-Order Differentiability of the flow of a System with LP Controls, J. Optim. Theory Appi, 89, 293-309 (1996). 11. Bianchini, R.M. , Good needle-like variations, in Proceedings of Symposia in Pure Mathematics, 64, 91-101 (1999). 12. Sussmann, H. , Lie Brackets and Local Controllability: a Sufficient Condition for Single Input system, SIAM J. Control Optim., 2 1 , 686-713 (1983). 13. Bianchini, R.M. , and Stefani, G. , Graded Approximations and Controllability Along a Trajectory, SIAM Journal on Control and Optimization, 28, 903-924 (1990). 14. Bianchini, R.M., High Order Optimality Conditions, to appear in Rendiconti del seminario matematico di Torino.

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233

GEOMETRIC OPTIMAL CONTROL OF T H E A T M O S P H E R I C A R C FOR A SPACE SHUTTLE B. BONNARD,E. BUSVELLE, AND G. LAUNAY Laboratoire d'Analyse Appliquee et d'Optimisation Universite de Bourgogne, UFR des Sciences, 21000 Dijon, France E-mail: [email protected], [email protected], We give preliminary remarks concerning the optimal control of the atmospheric arc for a space shuttle (earth re-entry or Mars sample return project). The system governing the trajectories is 6-dimensional, the control is the bank angle, the costintegrand is the thermal flux and we have state constraints on the thermal flux and the normal acceleration. Our study is geometric and founded on the analysis of the solutions of a minimum principle and direct evaluation of the small-time reachable set for the problem taking into account the state constraints.

1

Introduction

The objective of this article is to make a preliminary analysis of the optimal control of the atmospheric arc for a space shuttle where the cost is the total thermal flux. The control is the bank angle (the angle of attack being hold fixed) and we have state constraints on the thermal flux and the normal acceleration. A pure numerical approach to the problem is presented in 2 where the analysis is also simplified because the terminal condition is relaxed to a condition on the modulus of the speed. Our aim is to analyze the problem with fixed end-point conditions which leads to a complex control law due to the number of switchings (or the number of rotations) we need to match the boundary conditions. This article is only a first step in the analysis in order to introduce the geometric tools to handle the problem and the necessary optimality conditions. In particular we shall restrict our computations to a 3 dimensional subsystem where the state variables are the modulus of the velocity, the altitude and the flight path angle. Also we shall localize the analysis to a small neighborhood of any point in the flight domain. This will allows to give local bounds to the number of switchings. It must be completed by numerical simulations to get a global bound. Our approach is geometric and use necessary optimality conditions and direct evaluation of the small time reachable set in the spirit of n but using normal forms as in 2 where the constraints are taken into account. It is well illustrated by the following planar example. Consider the time optimal

234

control problem for the system q = X (q) + uY (q), q = (x,y), \u\ < 1. Let 7+ (resp. 7_) be an arc corresponding to u — + 1 (resp. u = — 1) and denote 7172 an arc 71 followed by 72. Take a generic point q0, then the small time reachable set starting from q0 is a cone bounded by arcs 7+ and 7_ and each optimal trajectory is of the form 7 + 7 - or 7_7+, see figure 1; moreover along a trajectory the time can be measured using Miele's form: UJ = pdq where p is given by (p,X) = 1, (p,Y) = 0.

Figure 1. Reachable set with and without constraints

Assume qo = 0 and the trajectories constrained to the domain C : y > 0. Let 76 (t), t € [0,T] be a boundary arc starting from q0 = 0 and contained in the frontier y — 0; assume that the corresponding control Ub is admissible and not saturating. Let B — 7;, (£), T > 0 small enough. Consider the arcs 7+7_ and 7-7+ joining 0 to B , one is time minimal (and the other is time maximal) for the problem without state constraint, and we have two possibilities for the constrained problem, see figure l,(b). Assume it is 7 + 7 - , then if it is contained in y > 0, it is admissible and the boundary arc is not optimal, the optimal synthesis near qo for the constrained system being 7 + 7 - . If 7 + 7 - is not contained in y > 0 the boundary arc is time optimal and the optimal synthesis is 7 + 7(,7_. The analysis can be carried out in full details using the model x = 1 + ay, y = c + u and not the Miele's form w defined only for planar systems.

235

A major problem when analyzing optimal control problems with state constraints is to derive necessary optimality conditions. Indeed the constraints can be penalized in the cost in several manners and this leads to introduce the concept of order of the constraints. Also it is the basic concept to construct normal forms and evaluate the reachable sets for the system with the constraints. We shall formulate a minimum principle due to 8 ' 1 2 , adapted to analyze the optimal trajectories for the space shuttle. It concerns single input control systems and we need regularity assumptions. It is much more precise than the general minimum principle of 12 , where an optimal trajectory is the projection of a trajectory in cotangent bundle depending of a measure supported by the constraints. 2

The model

The problem is to control the atmospheric arc nearby a planet which can be the Earth (re-entry problem) or Mars (sample return project). In both cases the equations are the same, except for constants related to the planet (radius, mass, angular velocity, atmosphere). In our computations we shall assume that the planet is the Earth. In order to modelize the problem, we use the laws of classical mechanics, a model of the gravitational force, a model of atmosphere and a model of the aerodynamic force which decomposes into a drag force and a lift force. The equations are simplified by choices of orthonormal moving frames that we explain below. 2.1

Moving frames

We denote by E = (ej,e2,e3) a standard Galilean frame whose origin O is the center of the Earth and let i?i = (/, J, K) be a rotating frame centered at 0 where K is the axis N-S of rotation of the Earth, the angular velocity being f2 and I is chosen to intersect Greenwich meridian. Let R be the Earth radius and let G be the center of mass of the shuttle. We denote by R[ = (e r ,e(,e£,) the frame associated to spherical coordinates of G = (r,l,L), r > R being the distance OG and /, L being respectively the longitude and latitude. We introduce the following moving frame R2 = {i,j, k) whose center is G. Let C : t -> (x (t), y (t), z (t)) be the trajectory of G measured in the frame Ri and let ~& be the relative speed v = xl + y J + zK. To define i , we set ~& = \v\ i . The vector j is a vector in the plane (i, er), j is perpendicular to i and oriented by j.er > 0. We take k = i A j . The vector i is parametrized

236

(a)

(b)

Figure 2. Moving frames: flight path angle and azimuth

in the frame R[ = (er,ei,ei)

by two angles:

• 7: flight path angle • E: azimuth defined on figure 2. 2.2

Model of the forces

For the atmospheric arc we assume the following Assumption 1 There is no thrust: the shuttle is a glider. Assumption 2 The speed of the atmosphere is the speed of the Earth, i.e. the relative speed of the shuttle with respect to the atmosphere is the speed it. We must consider two types of forces acting on the shuttle. • Gravitational force. We assume that the Earth is spherical so that the gravitational force is oriented along e r .It is written in the moving frame R2 P = — mg (i sin 7 + j cos 7) where g = % .

237

• Aerodynamic force. The effect of the atmosphere on the shuttle is on aerodynamic force which decomposes into

— A drag force colinear to the speed ~& and of the form

2* = - (^pSCDvA i

— A lift force perpendicular to ~& and given by

~P T = -pSCiv2

(j cos p + k sin p)

and p is called the bank angle, where p — p{r) is the atmospheric density, S is a constant and CD, CL are respectively the drag and lift coefficient.

Assumption 3 Both coefficients Co and CL are depending upon the angle of attack a which parametrized the orientation of the speed v with respect to the normal of an element of area of the shuttle. We assume that for the atmospheric arc the angle of attack is kept constant. This is very restrictive but it is worth to point out that in the numerical simulations of 2 where a is a control, in the optimal solution it is a constant. Hence the only control is the angle of bank p.

2.3

System equations

The atmospheric arc is governed by the following system

238

dv — =t/sin(7) d v

•

(2.1a) !

I \

$ CD

o

— = —g sin (7) — -p .£

C*E

n.2

r , •

T

•

T

v + s i r cos L (sin 7 cos L - cos 7 sin L cos s ) TIL

(2.1b) — = cos (7) ( at

\ 2

+ fl dL

v

1- - ) + -zP v

r>

2

wcos(u) + 2ficosLsinE;

(2.1c)

m

(cos 7 cos L + sin 7 sin L cos E) „

(2-ld) ,

-7- = - cos 7 cos = (2.1e) (2.1f) at r dl v cos 7 sin H d=. 1 5(7^ u v , . _ . , x — — = -/> 1—cos 7 tan L sin ^ (2.1g) dt r cossmu L dt 2 m cos 7 r „~ / - ,—s ^9 r sin i cos L sin 5 ,„ , . + 2H (sin L — tan 7 cos L cos ,=,) + f r (2.1h) v cos 7 where the control is the bank angle fi and the state space is q = (r, u, 7, L, I, E) 2.4

Atmospheric model

Atmospheric density is tabulated for Earth, Mars and Venus and we take an exponential model P = Poe~0r 3 3.1

The control problem Control and control bounds

The control can be either /x or /t. In the first case we can have the following bounds: /J, £ [— | , | ] or ft 6 [—7r,7r]. We set Ui = cos/x and «i is a direct control on the flight path angle 7. We let «2 = sin^j and U2 control the azimuth, the sign of u?, allows the glider to turn left or right. 3.2

State constraints.

There are several state constraints but in the first step of our analysis we shall consider two constraints:

239

• Constraint on the thermal flux

< y>max

(3.1)

where Cq is a given constant • Constraint on the normal acceleration 7n=7n0(«)^2<7rx 3.3

(3-2)

Optimal cost

Several choices are allowed and we make the analysis for T

J ( / i ) = f Cq^v3dt (3.3) Jo which represents the total thermal flux, the duration T of the atmospheric arc being not fixed. We introduce the differential equation ^

= C , ^

s

(3-4)

with go (0) = 0. 3.4

Boundary conditions

The transfer time T is free and we have two choices for the boundary conditions: • Fixed boundary conditions at t — 0 and t = T for q —

(r,v,j,L,l,E).

• 7 £ [7min> 7max] at t = 0 with the constraint that a preliminary maneuver on the Keplerian arc allows this possibility. 3.5

State domain for the atmospheric arc

The flight domain D for the Earth re-entry of the shuttle is the following: • Altitude: h = r - R G [40 km, 120 km] • Velocity amplitude v 6 [2000 m / s , 8000 m / s ] • The flight path angle domain i s 0 > 7 > — 1 5 ° .

240

Assumption 4 (controllability assumption) The Earth angular velocity ft is small and hence

|-«(7)(-|

+

J) + i^.«-M

(3.5)

We shall denote by Dc the subset of D where the lift force can at each point compensated the gravitational force that is 1 SCL g -p v > 2 m v and (3.5) is feedback linearizable in the domain. 4

The minimal principle without state constraints — Extremal curves

4-1

Problem statement and notations

Let the single-input control system q = F(q,u)

(4.1)

and a cost to be minimized of the form J (u) = / if (q) dt (4.2) Jo where the transfer time T is free and tp is not depending upon u. The set of admissible controls is the set U of measurable mappings u : [0, T] —> U. The state domain is a subset of W1 with the state constraints: • Constraint on the thermal flux ci (q) =
(4.3)

• Constraint on the normal acceleration C2 (q) = In (?) < oa

(4.4)

The boundary conditions are of the form: • q (0) =
then 7(0) G [71,72], 7i < 72 < 0.

We denote by R (q0, t) the reachable set at time t > 0 fixed and R (q0) = U small enough R (*>, *) the small time reachable set.

241

4-2

Minimum principle

We recall the minimum principle 13 which allows to parametrize the boundaries of the reachable sets n . We introduce the Hamiltonian H (q, p, u) = {p,F(q, u)) + pQtp {q) where q = (r,v,j,L,l,E) and p = (pr,Pv,P-y,PL,PhPs) is the adjoint vector and po is a constant such that (p,po) j1 0. Definition 4.1 If po ^ 0 we are in the normal case and ifpo =0 we are in the abnormal case. Definition 4.2 We call extremal a triplet (q,p,u) solution of the minimum principle q

= F(q,u)

dF _ dip - ^q-- °-dq= min H (q,p,w)

P= p

H{q,p,u)

= —

p

(4.5) =

8H -dq-

^ (4.7)

wEU

Proposition 4.3 An optimal solution for the problem without state constraint is a projection on the state space of an extremal solution. Moreover po > 0. Since the transfer time T is free it is exceptional, that is H = 0. If moreover 7 is free at t = 0, the adjoint vector p satisfy the transversality condition p7(0)=0i/7(0)€]7l,72[ 4-3

(4.8)

Definition of subsystem (I)

Observe that fi is small with respect to the velocity of the shuttle. Hence if we neglect the transport terms O (fi2) and the Coriolis terms O (Q) our system can be decomposed with q\ = (r, ^,7) and qi — (L, i,H) into Qi = Fi

(9i>"i)

where u\ — cos/z, w2 = sin^, u = (1(1,^2) and u

= \u\ + ul - 1 a n d ui > 0 if /a G ~, ^1 j

242

The adjoint system (4.6) is the decomposed into

I f 0 0\ {PlP2 0)=-{PlP2Po)

| |§f fff 0

fe 0 0 If we relax the end-point condition on q2 = (£,'>£) we obtain using the transversality condition p2 (T) = 0 and hence p2 (t) = 0. The analysis of extremals reduces to the analysis of the solutions of 9i =

Fi(q1,u1) dFi _ dip

Pi = -Pi-E oqi

Po^— dqi

It is associated to the optimal control of system (I) given below: dr (I)

= v sin (7)

dv . . . 1 SCD 1 — =-gsm{rt)--p——v at 1 m d-y . . / g v\ 1 SCL ,N — = c o s ( 7 ) ( - - + - ) + - p — — «cos(/i)

Note that q\ = (r, ^,7) appears only in the state-constraints. We shall concentrate in a first step our analysis on the subsystem (I). It is related to the numerical simulation of 2 . 4-4

Analysis of extremals of system (I)

4-4-1- Problem reduction and definitions Consider a single input affine system q = X + uY,

\u\<\

(4.9)

and a cost to be minimized of the form J ( « ) = /
(4.10)

Assume moreover that tp(q) > 0 in the state domain. Introduce the equation / % =¥> (q) X 9o (0) = 0

243

and q = (q,qo) is the enlarged state. Hence (4.9), (4.10) can be written q = X(q) + uY(q),\u\
(4.11)

and let s be the new time parameter defined by ds =
(4.12)

and if q' denote the derivative of q with respect to s, (4.9) can be written q' = X(q) + uY(q),\u\
(4.13)

where X = ipX, Y = tpY and ip = -. The optimal control problem becomes a time minimum control problem. Definition 4.4 Consider q = X + uY. A singular trajectory of the system (X, Y) is a projection of the following equations dH q = dp dH i> = dq uq (p,Y)=0

(4.14)

where H = (p,X + uY), p ^ 0. It is called exceptional if H = 0 and, admissible if \u\ < 1 and strictly admissible if u £ ] —1, +1[. Notation 1 IfXi and X2 are two smooth vector fields, we denote by [Xi,X2] the Lie bracket computed with the convention [X, ,X2](q) = ^

(q) X, (q) - ^

(q) X2 (q)

Assumption 5 Throughout this article we shall assume that g = ^ is constant. Proposition 4.5 In the domain cos 7 7^ 0 there is no exceptional singular arc for the system (X, Y). Proof. The singular extremals are located on (p, Y (q)) = 0. Differentiating twice with respect to t one gets (p,[X,Y](q)) = 0 (p, [X, [X, Y}} (q)) + u{p, [Y, [X, Y]] ( = 0

244

We must compute the Lie brackets Y, [X,Y], [Y,[X,Y]] and [X,[X,F]] where . d , . , o\ d / g v\ d X = vsin7(gsin7 + kpv ) — + cos 7 h - TTv or ov \ v r) 07

and k , k are defined by the equations (2.1b) and (2.1c). Since the concept of singular arc is feedback invariant we can replace in our computations X and Fby v . d X = vsin7or

,

.

v a sin

, 2, d 7 + kpv ) — ' ov

We have then [X,Y\ = - t ; c o s 7 — +g cos 7 — [r, [X, y]] = v sin 7 — - 5 sin 7 — hence [X, Y] and [y, [X, Y]] are colinear. Moreover [X, [X, Y]] = kpv2 cos 7-5- + (-kv3p' or

cos7 + 2fcpt/t; cos7) — ov

The singular extremals are located on £': {p, Y) = {p, [X, Y]) = 0 that is P-r = Pv5 - prv = 0. We introduce D = D' = D" =

det(Y,[X,Y],[Y,[X,Y}}) det(Y,[X,Y),[X,[X,Y}]) det{Y,[X,Y],X)

From our previous computations singular arcs are located on D = D' = 0 and moreover if they are exceptional they satisfy D" = 0. We have £> = 0 D' = kv2 cos2 j (p v2 — 3pg) D" = kpv3 cos 7 Since C0S7 7^ 0 the proposition is proved. Moreover we have for system (4.9)

D

245

Lemma 4.6 If cos^f ^ 0 then 1. Y and \X, Y] are independent; 2.

[Y,[X,Y}]eSpan{Y,[X,Y]}

4-4-%- Analysis of extremals Consider the time minimum control problem for system (4.13): q'=X(q)+uY(q),

|u|
We introduce the following definitions Definition 4.7 The set S : {p,Y (q)) = 0 is called the switching surface. Let (q,p,u) be an extremal defined on [0, T]; it is called singular if it is contained in S, bang if u — + 1 or u = — 1 and bang-bang if u(t) is piecewise constant and given a.e. by u(t) = — sign (p (t), Y (q (t))). We denote respectively by 7+ (resp. 7_, 7 S ) a smooth arc associated to u = + 1 (resp. u = —1, u singular control) and 7 + 7 - represents an arc j + followed by an arc 7_. Let us calculate Lie brackets. We have , ( • d , . , 9, d X = i> ^ s i n 7 ^ : - (<7Sin7 + kpv)fo+

/ g v\ d \ c o s 7 (-- + -) g^J

-*F

Y = ipkpv— 07 where tp = ip~l. Since X = ipX, Y = ipY using for fi, J2 smooth functions the formula [hX, f2Y] = fxf2 [X, Y] + h (Xf2) Y-h

(Yh)

X

where Zf = | £ z (q) is the Lie derivative we get \X,Y]

=*l>2[X,Y] +

i>(Xilj)Y-iP(YiP)X

Since Y = kpv-^- and ip = f(p,v) we have Yip — 0. Hence \X, Y] — ip2 [X, Y]+I/J (Xi/)) Y. Computing [F, [X,F]] as before we obtain Lemma 4.8 1. The set £ ' : (p,Y)
= f/f3\Y,[X,Y]] €Span{r,[X,y]}.

= (p, \X,Y]) mod

= 0 is given by (p,Y) =

Span{F, [X,Y]}

and

hence

246 Moreover,

D = det(F,[X,F],[F,[x,F]])=o and P" = det(F,[X,F],X)

=

^

I

and hence Y, [ X , F ] , X are a frame in the flight domain where cos 7 7^ 0. So there exists a, b, c such that \X, \X,Y]]

= a~X + bY + c \X,Y]

(4.15)

Long computations give us the crucial result Lemma 4.9 If cos 7 ^ 0 we have 1. D' = det ( F , [ X , F ] , [X, [ X , F ] ] ) = - | ^ - c o s

2

7 *0

Corollary 4.10 If cos 7 7^ 0 t/sere exists no singular trajectory. 4-4-3. Application to the classification of extremals near E Let (q,p,u) be a smooth extremal on [0,T]. Differentiating the switching function $ : t -> (p (t), Y (g (£))) we have

*(t) = (p(t),[x,F] (?(*))> $(t) = (p(t), {X, \X,Y\] (q(t))+u(t) [F, [X,F]] (g(t))> We use the results of 10 to classify the extremals near a point ZQ = (qo,Po)1. Ordinary points. If z0 belong to (p, Y) = 0, (p, [ X , F ] ) 7^ 0, the point 00 is called of order 1 or ordinary and each extremal curve is locally of the form 7 + 7 - or 7-7+. 2. Points of order 2. Let z0 E £': (p,Y) = (p,\X,Y])= 0. Then if (q,p,u) is a smooth extremal through ZQ the switching function satisfies at ZQ: *(*) = $ (t) = 0

247

and * ( t ) = (p(t), =

\X, \X,Y]]

(q(t))+u(t)

[F, \X,Y\]

(«(*)))

(p(t),\X,{X,Y]](q(t)))

from lemma 4.8 which is non zero from lemma 4.9. Hence both curves corresponding to u — + 1 and u = — 1 have a contact of order 2 with respect to £ and the extremal solutions are represented on figure 3. Ac-

Figure 3. extremal solutions (a > 0)

cording to the classification of 10 the point ZQ is a parabolic point and each extremal is locally bang-bang and of the form 7 + 7 _ 7 + or 7 _ 7 + 7 _ . Prom this analysis and from the minimum principle we can conclude about small time optimal policy. Theorem 4.11 If cos 7 ^ 0 each small time optimal policy is of the form 7_7+7_ where 7+ is an arc corresponding to u = cos/z = + 1 and 7_ an arc corresponding to u = — 1 (or u — 0 if n £ [— | , | ] ) . Proof. __ According to the time minimum principle, an optimal arc has to satisfy H — 0, p0 > 0 where H — {p,X (q) + uY (q)) + p0 = 0. Hence p can be oriented at z0 € £ ' according to (p,X (q)) < 0. Write [X, \X,Y]]

=alC + bY + c

\X,Y]

248

and a < 0 from lemma 4.9. Hence from figure 3, only an extremal 7_7 + 7_ can be optimal. The assertion is proved. • 4-5

Geometry of the small time reachable set

Consider again system in 3-dimension

^=X(q)+uY(q) and its time extension in 4-dimension by adding the cost ^ j - = 1. We denote respectively by R(qo) the small time reachable set and by R(qo,0) the small time reachable set for the extended system. One major research program undertake in 1 4 ' n using original ideas from Lobry is to evaluate in small dimensions the small time reachable set and its boundary. In particular the following result is basic: Lemma 4.12 Consider system (X,Y) in dimension 3 and let gi = X + Y and g2 = X — Y. Assume g\, g2 and [#1,(72] linearly independent at qo then R (qo) is bounded by the two surfaces 7 + 7 - (go) and 7-7+ (go) and moreover

R(qo) = U 7+7-7+ (go) (or U 7-7+7- {qo)) Actually in theorem 4.11 we proved more (see also 1 4 ) : Lemma 4.13 If cos 7 ^ 0 the boundary of the small time reachable set for the extended system R (qo, 0) is an union ofj-j+ 7_ (go, 0) and 7 + 7 _ 7 + (g 0 ,0) where 7 denotes the time extended trajectory. 4-6

Optimal control of the atmospheric arc

If we consider the complete set of equations it can be written as a time optimal control problem for a 6-dimension system of the form

q'=X(q)+u1Y1(q)+u2Y2(q) where 111 = cosju and u2 = sin/i. We can use two points of view related to the control device: • We set jj, = w where w is taken as a control bounded by M. The system is then a single input affine control system on the 7-dimensional state space (g,£t). • We can consider the original system on the 6-dimensional state space. The control u = (MI,W 2 ) satisfies u\ + u\ = l. If fi € [0,27r], the optimal control problem is equivalent to a sub-Riemannian problem with drift.

249 Indeed if we set ipi = (p, Yi{q)), given by u= ^ ( ^ 1 , ^ 2 ) -

i = 1,2 an extremal normal control is

We must analyze the existence of abnormal extremals and the number of oscillations and switchings of optimal trajectories. 4-7

Conclusion about this section

Using minimum principle, Lie brackets and geometric methods we have evaluated the small time reachable set and solved the small time optimal control problem for system I. In particular we have obtained bounds on the number of switchings. The main property of system I is that [Y, [ X , F ] ] belong to Span {Y, [X, Y]}. This is connected to the feedback linearizability if system I. For the global aspect one needs to analyze the global proportion of the switchings function $ using convexity analysis and Rolle theorem. Our study is a preliminary step in order to evaluate the reachable set for the full system of equation without the state constraints and nearby the state constraints. We shall analyze the structure of the reachable set for system nearby the constraints in the next section. 5

Optimal control with state constraints

In this section we analyze the optimal control problem for system I, taking into account the constraints. We recall a minimum principle from 12 adapted to our situation. Our contribution is to make a direct evaluation of the small time reachable set for the constrained system using the previous computations of section 4 and a normal form. When dealing with constrained systems the main concept is the concept of order of the constraint that we define next before to state the minimum principle adapted to our analysis. 5.1

A minimum

principle

We consider the single input affine control system q = f{q)+ug

{q)

\u\ < 1

and a cost to be minimized of the form

J(u)=G{x{T)) where the transfer time T is fixed and q is constrained to c(q)<0

250

The boundary conditions are g(0) = go $ [x (T)) = 0 The problem is denoted by (Vo) and can be imbedded into the one parameter family of problems (Va) where the constraints set is taken as c

(Q) < a, a small The important concept is the concept of order of the constraint. Definition 5.1 The absolute (or generic) order of the constraint is the first integer n + 1 such that g{foc)=g{flc)=---=g(fn-1c)=0

fl(/nc)#0 where the vector Definition 5.2 tained in c — 0. by differentiating

fields f, g acts on c by Lie derivative. A boundary arc t >-> jb (t) is a solution of the system con/ / the constraint is of order n it can be generically computed n times the constraint and solving the linear equation c<"> = fnc + ug {r~lc)

= 0

(5.1)

A boundary arc is contained in c = c=-=

c(n_1) = 0

(5.2)

-1

and the constraints c = • • • = c'™ ' = 0 are called the secondary constraints. We denote by Ub the feedback control rSL f ^ which allows to remain in the constrains set. Let's now formulate the Maurer minimum principle 12 : Assumption 6 (General assumption) We assume that the following conditions hold on a boundary arc s H-> 71, (s), s € [0, t}: (He) g (/ r a _ 1 c) |7i,7^ 0 (n being the order) (H7) |w&(£)| < 1 i.e. the boundary feedback control is admissible and not saturating. Necessary conditions. Define the Hamiltonian by H(q,u,p,n)

= (p,f + ug)+nc

(5.3)

where 77 is a Lagrange multiplier of the constraint set. The necessary conditions of the minimum principle are the following:

251

Condition 1 1. There exists r}(t) > 0, a real number r]o > 0 and 5 such that the adjoint (row) vector satisfies

p(T)=Vo^(q(T))+6~(x(T))

(5.5)

2. The function r) (t) satisfies n (t) c (q (t)) = 0 Vi € [0, T] and is continuous on the interior of the boundary arc. 3. The jump condition at a contact point or a junction time t\ is Br P (tf) = P (*r) - "i Q- (9 (*i)), "i > 0 4- The optimal control u(t) minimizes the Hamiltonian, H (q(t) ,u(t) ,p(t) ,ij(«)) = min H (q(t) ,u,p(t)

(5.6)

i.e. ,n(t))

(5.7)

|u|
Remark 5.3 In this minimum principle, only the constraint c is penalized in H; others choices are possible using the secondary constraints, see 8 ' 1 3 . Remark 5.4 There exist a general minimum principle without assumption (He), see for instance 9 where the adjoint equation (5-4) takes the form

P(«) = - / P ( » ) ( ^ + « | ) * - / | ^ where \ii is a measure supported on the set c = 0. Our principle is much more precise because from (5.4) the measure is of the form dfii = T) (t) dt where n is C°. This additional regularity comes from assumption (HQ) and at non generic point where g (fn~1c) vanishes n can blow up. The case where T is not fixed can be deduced from the case where T is fixed. We introduce a new variable z — T and the system dt _ ds

fs=(f(q)+ug(q))z

252

We have s = £ and the trajectories are parametrized by s € [0,1]. The new transfer time is 1. An important research program is to analyze the solutions of the minimum principle with constraints. This analysis is outlined in 12 . An interesting point of view is to analyze the open loop solution deduced from the problem without constraints by analyzing the bifurcation of an unconstrained optimal solution when the constraint c (g) < a becomes active. Next we adopt a different approach based on the evaluation of the small time reachable set near the constraints. It will provide necessary and sufficient optimality conditions. 5.2

A direct approach

5.2.1. Order of the constraint Consider in the shuttle problem the constraint on thermal flux ci = Cgy/pv3
poe~0r

= 0 is a secondary constraint. Moreover

(r,v)cos7 where
Similarly for the normal acceleration c2 = 7n 0 p^ 2 we get ci = -7n 0 (2fy>V + {Ppvz + 2gpv) sin 7) i.e. c2 = iphir,v) + tp6 (r, v) sin 7 = 0,
253

constraint is of order 2, hence Yc = 0 near 0 and Y is tangent to all the surfaces c = a. Hence | | = 0. • Normalization 2. Since c is not depending upon z using a diffeomorphism preserving Y ~ ^ we can normalize c t o c ( x , j / ) = x. The system can be written x = Xi (q) V = X2 (q) z X3(q)+u and c = x. The secondary constraint is x = 0 and we assume that x = x = 0 is an arc a passing through qo = 0- If we keep the affine approximation sufficient for our analysis we obtain a system which can be written x = aix + a2y + a3z y = b0 + hx + b2y + b3z z = co + c\x + c2y + c3z + u where a is approximated by the straight line x = a2y + a3z. If bo ^ 0 (generic case) we can assume bo = 1. • Normalization 3. Changing z into — z and u into — u if necessary and using a transformation of the form Z = ay + z one can identify a to x = z — 0 and the system can be written x = aix + a3z y=l + b1x + b2y + b3z z = Co + C\X + c2y + c3z + u

(5.8)

where a 3 > 0. If moreover the boundary arc is admissible and not saturating (assumption (H7)) we have the condition |co| < 1. Theorem 5.6 Consider the problem of time minimization in q = X (q) + U Y (?)j ? € M3 subject to c (q) < 0. Let q0 £ {c = 0} and assume the following:

1. Nearqo, [F, \X,Y\] € Span{F, \X,Y]} 2. X,Y,

\3(,Y]

are linearly independent at qo and

\X, \X,Y]] with a < 0

(q0) = aX(q0) + bY (q0) + c{X,Y]

(q0)

254

3. The constraint c = 0 is of order 2 and assumption fH6/) and (H7) are satisfied at qo then the boundary arc through qo is small-time optimal if and only if 7_ (qo) is contained in the domain c > 0. Proof. From lemma 4.12, we know that each small time reachable point from qo can be reached by an arc 7+7-7+ and 7 - 7 + 7 - and from theorem 4.11 we know that the small time optimal arc is of the form 7 - 7 + 7 - for the unconstrained system. Let the constrained system written as (5.8) in the normal coordinates where q0 = 0 and the boundary arc •jb (t) is identified to (0, t, 0). Let B = 76 (t), t > 0 small enough. Let u = + 1 or u = —1. For a trajectory with q (0) = 0 we have the following approximations: z (t) = (co + u) t + o (t) t2 x (t) = a3 (co + u) — + o (t) Hence the projections of the arcs 7+7-7+ and 7 - 7 + 7 - joining 0 to B in the plane (x,z) are loops denoted 7+7-7+ and 7 - 7 + 7 - and are represented on figure 4.

Figure 4. Projection of the arcs 7+7-7+ and 7 - 7 + 7 -

In particular, we proved the following.

255

Lemma 5.7 The loops 7 - 7 + 7 - (resp. 7+7-7+,) are contained in the domain x < 0 (resp. x > 0). We can now end the proof of the theorem (the assertions concern system

(X,Y)). If the arc 7 - 7 + 7 - to join 0 to B is contained in the domain c < 0 it is time minimal and the boundary arc is not optimal. If the arc 7 - 7 + 7 - is contained in c > 0 then we can join 0 to B by an arc 7+7-7+ in c < 0. But the analysis of section 4 replacing min t by max t shows that such an arc is time maximal. Hence a bang-bang arc 7+7-7+ in the domain c < 0 joining 0 to B cannot be optimal. Then the boundary arc 76 is optimal. • Moreover Corollary 5.8 / / a boundary arc % is small time optimal then there exist optimal trajectories of the form 7-7+767+7-. 5.2.3. Application to the shuttle For the shuttle we have a < 0, so we have to consider loops 7 - 7 + 7 - where 7_ corresponds to cos fi = 0 or cos fj, = — 1. From the computations of section 5.2 we have for both constraints c\, c-i: '6i = $ (r, v, 7) + u cos 7 $ (r, v) where $ < 0. Hence c, is minimal when /i = 0 ° i.e. u — + 1 . Assume that the parameters of the problem are such that assumption (H7) is satisfied. Then the arcs 7_ through the boundary points are contained in the non admissible domain and the boundary arc is optimal. 6

Conclusion

We have outlined the geometric research program to analyze the optimal control of the atmospheric arc for the space shuttle. Our tools are necessary optimality conditions and evaluation of the small time reachable set. Near the constraints the evaluation is related to the classification of pairs of vector fields near a surface. This problem is common to several problems met in optimal control: classification of extremals near the switching surface, optimal control with targets and so on. References 1. O. Bolza, Calculus of variations (Chelsea, 1973).

256

2. F. Bonnans and G. Launay, Large scale direct optimal control applied to the re-entry problem, Journal of guidance, control and dynamics, 2 1 , 6, 996-1000 (1998). 3. B. Bonnard, G. Launay, Time minimal control of batch reactors, COCVESAIM, 3, 407-467 (1998). 4. B. Bonnard and I. Kupka, Theorie des singularites de l'application entreesortie et optimalite des singulieres, Forum Math., 5, 11-159 (1993). 5. A. E. Bryson and Y. C. Ho, Applied optimal control (Hemisphere Pub. Corporation, 1975.) 6. CNES, Mecanique spatiale, tome 1/2 (Cepadues Eds., 1995). 7. J.-M. Coron and L. Praly, Guidage en rentree atmospherique, Rapports 4/5 (CNES, Octobre 2000). 8. D. H. Jacobson et al., New necessary conditions of optimality for control problems with state-variable inequality constraints, J. Math and Appl., 35, 255-284 (1971). 9. A. D. Ioffe and V. M. Tikhomirov, Theory of extremal problems (North Holland, 1979). 10. I. Kupka, Geometric theory of extremals in optimal control problems, TAMS, 299, 225-243 (1973). 11. A. J. Krener and H. Schattler, The structure of small time reachable sets in small dimensions, SIAM J. on Control and Op., 27, 120-147 (1989). 12. H. Maurer, On optimal control problems with bounded state variables and control appearing linearly, SIAM J. Control Optimization, 15, 345362 (1977). 13. V. Pontriaguine et al, Methodes mathematiques des processus optimaux (Ed. MIR, 1974). 14. J. H. Schattler, The local structure of time optimal trajectories in dim. 3 under generic conditions, SIAM J. Control Optimization, 26, 4, 899-918 (1988). 15. H. Sussmann, The structure of time-optimal trajectories for single input systems in the plane: the C°° non singular case, SIAM J. Control Opt., 25, 856-905 (1987).

257

H I G H - G A I N A N D N O N - H I G H - G A I N OBSERVERS FOR N O N L I N E A R SYSTEMS E. BUSVELLE A N D J.P. G A U T H I E R Departement de Mathematiques, Universite de Bourgogne, Laboratoire d'Analyse Appliquee et Optimisation, BP 47870, 21078, Dijon cedex, France E-mail: [email protected] In this paper, following ideas already developped in 1 0 , we construct an observer for nonlinear systems that looks like the extended Kalman filter. In fact, it is asymptotically (in time) exactly the deterministic version of the extended Kalman filter, and when the "innovation" is large, it is an high gain observer. In the context of the theory developped in 1 0 , we show that it works for "all observable systems". In the paper, we prove convergence of the estimation error, we give several estimates of this error, and we show a convincing illustrative application (a distillation column)

D e d i c a t e d t o Velimir Jurdjevic

1 1.1

Introduction, systems under consideration Systems under consideration

We consider nonlinear systems of the following form (1.1), on Rn. The control space U, is a closed subset of Rd. Only for simplicity of the exposition of the proof of the main result, the observation is taken to be single-valued: it is a u— dependant linear form on Rn. dx — =A(u)x UT

+ b(x,u),

y = C(u)x. A(u) , C(u) are matrices: C(«) = (oi(u),0,....,0),

(1.1)

258

( 0,a 2 (u),0,....,0 \ 0,0,a 3 (u),0,...,0 A(u) 0,

,0,a n (u)

V°.

»o/

where aj(.), i = l,...,n, are positive smooth functions, bounded from above and from below: 0 < am
\-b2(x1,x2,u)OXi

\- ... + OX2

bn(x1,...,xn,u)-—. OXn

These assumptions look very strong. In fact, under either genericity hypotheses or observability hypotheses, for the purpose of synthesis of observers, it is sufficient t o restrict to these systems, under the normal form (1.1) (or similar multi-output normal forms), and meeting these assumptions. This will be discussed in the next section 2.

We stress again that in all the paper, the single output assumption can be removed everywhere, and we leave this to the reader, but, in Section 4, we will deal with a 2— outputs system, in a similar normal form. 1.2

Presentation of the paper

Our purpose herein is to construct observers, for the observable systems described above. In fact, for these systems, several types of nonlinear observers can be constructed. We will focus on two types of construction that both turn around the "extended Kalman filter", in either its deterministic or its stochastic form: 1. First construction: The Extended Kalman Filter itself, 2. Second construction: The High Gain Extended Kalman Filter, 3. Our construction in this paper: a mixing of 1. and 2. Let us just give some details now, to explain where we want to go.

259 1. The extended Kalman Filter. For long, the engineers introduced and successfully used the extended Kalman filter (EKF), either in its stochastic or its deterministic form. The EKF is just the standard Kalman filter for linear time-dependant systems, applied to the linearized system along the estimate trajectory. We will give precise equations later on. It is easy to see that it is a non-intrinsic object (depending on coordinates). It would be intrinsic if it was dealing with the linearized along the real trajectory, but this trajectory is unknown. It is known that, under observability conditions, the Extended Kalman filter, has good properties: (i) In its deterministic form, it is a local observer in the following sense. For sufficiently small initial error on the estimate of the state, the estimation error converges exponentially to zero. The prototype of these results can be found in 2 for instance. For our systems (1.1), with the assumptions of Section 1.1, it is not hard to check that the linearized systems along any trajectory are uniformly observable, (in the classical sense of the linear theory, and with uniform bounds on the Gramm observability matrices). Hence, this result applies. (ii) In its stochastic form, except for the linear case, where the EKF is the "optimal" filter, there is no general theoretical result that applies. Even for good observable systems in our normal form 1.1, for small noise, small initial variance and dimension 1: there is a counterexample of such a system, in 16 for instance, where the EKF doesn't work at all. Nevertheless, despite the lack of these theoretical justifications, people use it in practice for nonlinear filtering and it may give very good results (even for systems that have much weaker observability properties than those considered here). In the application of our techniques, presented in section 4 below, we will show a (family of) practical examples which is very interesting because, it seems that, the results of 16 on the EKF for small noise, apply in general, and that the "small parameter" has a physical interpretation. We will not say more about that because this is beyond the scope of this paper. But it is one more justification of the use of our method developed here to this application. 2. The High Gain Extended Kalman Filter. The following results have been proved in 4 , 5 , 10 .

260

We consider the equations of the extended Kalman filter, in which the "covariance matrix Q" depends on a real parameter 6,6 > 1, in the following way: Qij=Oi+i+1Qlr For 6 = 1, it is exactly the EKF. For 6 large enough, it is what we call here the "High gain extended Kalman filter" (HGEKF). (i) In the deterministic setting, the estimation error has arbitrarily large exponential decay (depending on 6). ( 10 , for instance). This holds whatever the initial error is, (that is, this is a global result). (ii) In the stochastic setting, it is a nonlinear filter with "bounded variance" (the variance is bounded in 6n, which is not that good, but it is bounded anyway). ( 4 , for instance). 3. What we want to do in this paper. The idea in this paper is the very simple following one: we give the parameter 6 in the HGEKF an exponential decay from 60 large, to 1. What is expected, (and what happens) is the following: (i) The beginning of the transient of the estimation error is the one of the high gain extended Kalman observer: there is an exponential decay that can be made arbitrarily large. (ii) There is a global exponential decay of the estimation error (but, of course, it cannot be controlled). (iii) The asymptotic behavior is the one of the standard "extended Kalman filter", (that people like in practice, as stated above). Our main result, Theorem 3.1 in Section 3 proves (i) and (ii). The proof is more or less an improvement of the proof of convergence of the high gain Kalman observer, as given in 10 . Of course, this construction has a terminal defect: it is time dependant. In deterministic terms, it will work for large initial estimation errors, but not for big "jumps" of the state at intermediate times. In the section 3.3, we propose a very simple practical way to make the observer "recursive". In the section 4, we show the application of this procedure to a binary distillation column in which the "quality of the feed" is unknown, an subject to large changes. It was already noticed in the book 10 that this application is a nontrivial nice application of the observability theory, and of high gain observers. Here, it is even much more convincing: when the feed changes, (a big "state jump"), the behavior of the observer is the one of a high-gain observer:

261

recovering arbitrarily fast the quality of the feed, and when the feed does not move, the asymptotic behavior of the observer is the one of the extended Kalman filter, almost optimal with respect to small noise in that case (but we do not prove anything about this optimality in this paper). For first applications of "high gain observers" to distillation columns, see 19

20

2

Justification of the assumptions and observability

2.1

Justification of the normal form

Let us recall here the main results of an observability theory summarized in 10 , and developed in 6 , 7 , 8 , 9 , 13 . This theory leads to the consideration of systems under the normal form (1.1), or similar multi-output normal forms such as (2.1) below, that meet the assumptions of the section 1.1. Here, by "observability", we mean "observability for every fixed input function u(t)n. For details, see 10 . The main results of this theory are as follows. They concern general nonlinear smooth systems of the form:

y=

h(x,u),

on a smooth manifold X, n dimensional, y £ W, u € U, subset of Rd. Basically, there are two cases. Case 1. p < d. In that case, observability is a non generic property. It is even a property of infinite codimension, at the level of germs of systems. This high degeneracy leads to the fact that, in the control affine case, all observable systems can be put under normal forms similar to (1.1). (moreover, one can take Oj = 1, i = l , . . . , n ) .

This result is only a local result in the state space, but it is a global result with respect to the control variable. Moreover, in most of the practical cases we know, it is also global in x. In particular, it will be global in the application of Section 4. In the non control affine case, there is another result, that we don't want to recall here. It leads naturally to high gain observers of another type ("Luenberger type"). Let us just say that the results herein can be easily generalized to this normal form and these observers.

262

Case 2. p > d. In that case, the situation is completely opposite. Observability becomes a generic property, and generically, a system can be put globally under a normal form similar to (1.1), but the dimension of the state in the normal form is bigger than the dimension of the state of the original system: it is at most double plus one. Also, the control in the normal form contains a certain number of derivatives of the control of the initial system. But this is more or less unimportant for observation problems, where the control, and hence its derivatives, are known. In fact, generically, the systems can be put in a form which is a very special case of the form (1.1), called the "phase variable representation":

y^=V{y,y,....JN-l\u,u,....MN-1)\

(2.1)

N < 2n + 1.

Other cases: there are also other (nongeneric) interesting cases where the original system can be put under the "phase variable representation" (2.1). For instance, systems without control that are such that the mapping: initial — sate —>• derivatives

of y :

{M)

x0^(y,y,....,y ), has "finite multiplicity" for a certain integer M. (See

10

, and originally

13

).

N o t e 1. The reasons for which we make the matrix A(u) depend on u in the normal form (1.1) may look not clear, because, in all the cases described above, it doesn't. In fact, the only reason to consider this dependance is the following: the formal computations we do in the proof of our main result, work for that type of systems. Moreover, in the application we describe in Section 4, the matrix A actually does depend on u. Note 2. In that case were Oj depends on u, the following should also be noticed: even the high gain version of the extended Kalman filter is much better in practice than the "high gain Luenberger observer" mentioned above: the high gain observers both kill the nonlinearities contained in the vector field b. But the extended Kalman filter takes into account the variations of u, through the matrix A(u). The standard high gain observers in Luenberger form don't do this. This is the case in the application, Section 4 below.

263

2.2

Justification of the technical assumptions

Let us consider successively the two technical assumptions we made in the section 1.1: A. 0 < am < a,i(u) < a,M, i = 1, •••, n, B. The functions bi are compactly supported. In fact, the assumption A is always satisfied in the cases 1., 2. of the previous section 2.1: the a; are constant and equal to 1. In the application of section 4, this assumption is also satisfied, as we shall see. Let us just notice the following. Al. The Assumption a,i(u) ^ 0 just implies observability of systems in the normal form (1.1): - If the output y(t) is known, the input being also known, the fact that a\(u) is nonzero implies that we can compute xi(t) from y(t). - The fact that 02 (u) ^ 0 implies that we can compute X2 (t) from the knowledge of xi(t), and by induction, we can reconstruct the whole state x(t) from the knowledge of y(t). Modulo a trivial change of variables, the condition a,i(u) ^ 0 is equivalent to a,i(u) > 0. A2. The aj being smooth, restricting to a compact subset of the set of values of control implies that we can find the am, OM, of assumption A. The assumption B above can be trivially realized, by multiplying by a cut-off function, compactly supported, leaving the original vector field b unchanged on an arbitrarily large compact subset of R". We cannot expect more than that. As explained in the book 10 , the problem of synthesis of observers is an ill-posed problem outside compact sets of the state space. This is easily understandable: on noncompact sets, it can happen that the estimation error goes to zero for certain metrics, but to infinity for others. So that, reasonable observers work only as long as the state trajectory x(t) of the system remains in a given compact set, or they work for semi trajectories {x(t),t > 0} that are entirely contained in a given compact set. To finish, let us mention that this restriction to compact sets (unavoidable in a general observation theory), has not so important consequences: for instance, the high gain observers can be used in general for global dynamic output stabilization (again, see 1 0 ) .

264

3

Statement and proof of the theoretical result

The observer we propose, is based upon the High gain extended Kalman filter, proposed in 10 , 4 , 5 . For computational details about the Riccati matrix equation, we refer to 3 , or 1 0 . 3.1

The observer and the statement of the theorem

The equation of the observer is: (i) f = A(u)z + b(z, u) - Sitter-1 (Cz - y(t)), ( « ) # = - U ( « ) + b*(z,u))'S - S(A(u) + b*(z,u))+ C'r-^C-SQeS,

^

1 J

f =A(l-0), where C = (oi(u),0,...,0) > Qe = 92\-1QA~\ A = diag(l, | , . . . , ( I ) " " 1 ) . Here, b*(z,u) denotes the Jacobian matrix of b(z, u) w.r.t. z, and r,X are positive scalars. Q is a symmetric positive definite matrix. Comments: 1. Q,r, in the stochastic context, are the covariances of the state noise and output noise respectively. 2. If A = 0 and 6Q = 1, or if A > 0, but t is large, this is exactly the (deterministic version of) the extended Kalman filter. 3. If #o is large, and if r < T, then, this equation is almost the equation of the high gain extended Kalman filter with gain 9(T). Hence, for T < T, setting e(-r) = Z(T) — X(T), (e is the estimation error), we can expect the following, for #o large enough in front of T: \HT)\\2

< e(T) 2 ( B - 1 >H(c)e-< O l '^- O 2 ) r ||e(0)|| 2 .

(3.2)

Here, 0,1,0,2 are positive constants, H(c) is a decreasing positive function of c, where 5(0) > c Id. Also, 9{T) = 1 + (0O - l)e~XT. In particular, this implies that the error e(t) can be made arbitrarily small, in arbitrarily short time, increasing #o- For 8 constant, this is the behavior of the "high gain extended Kalman filter. In that case (6 constant), this estimate follows from 1 0 , 5 . We will prove it below for 6 nonconstant. Our main result herein will be the following: Theorem 3.1. 1. For all 0 < A < Ao, (Xo = $^*2\' w^ere Q ^ Qmld and a comes from Lemma 3.2 below), for all 9Q large enough, depending on A, for all

265

SQ > c Id, for all K C IRn, K a compact subset, for all £o = ZQ — xo, £o € K, the following estimation holds, for all r > 0 : \\e(r)\\2 < R(X,c)e-a A(eo,T,\),

=

T

||e o || 2 A(0o,r,A),

2 n

+

(3.3)

XT

6o ( -V U-^-*- \

where a > 0. R(\, c) is a decreasing function of c. 2. Moreover the short term estimate (3.2) holds for allT > 0, r < T, for allOo > 0O, So = e A T ( ^ - l ) + l, where L'is the sup of the partial derivatives of b w.r.t. x. Comments. a. Note that the function A(0O,T, A) is a decreasing function of r, and that, for all r > 0, A > 0, A(60,T, A) can be made arbitrarily small, increasing 00-

b. This means that, provided that A is smaller than a certain constant Ao, and 0o is large in front of A, the estimation error goes exponentially to zero, and can be made arbitrarily small in arbitrary short time. c. The asymptotic behavior of the observer is the one of the extended Kalman filter, d. The "short term behavior" is the one of the "high gain extended Kalman filter". 3.2

Proof of Theorem 3.1

3.2.1 Preparation for the proof Let us recall that: 6(T) = 1 + (0O - l)e~XT,

(3.4)

and let us set F = diag(0,1,2, ...,n — 1). Then: d(|) _ dr

dA-1 dr

A(l - 0) 02 '

A(l-fl) 0

(3.5)

266

The equations under consideration are: (t) % = A(u)e + b(z, u) - b(x,u) - S W - ' C ' r ^ C e , ( « ) f = -(A(u) + b*(z,u))'S - S(A(u) + b*(z,u)) + C'r~lC -

SQeS, (3.6)

We make the following changes of variables, with P = S - l . x = Ax, z = Az, e = z - x, e = As, S = OA^SA'1, ? = §-'

= \APA,b{z)

= Ab(A-1z),b*(z)

=

(3.7) Ab*(A~1z)A-1.

u

Remark : It should be noted that the Lipschitz constant of b is the same as the one of b, and the maximum of ||6*|| is the same as the one of ||6*|| (recall that the component bi of b is compactly supported with respect to all of its arguments (xi,...,Xi,u), and that 6 > 1). An obvious computation gives: ±(e)

= 6[(A - PC'r-lC)e

l(S)

= 8[-{A + \l*[z) -(*-§+

+ \{b{z) - b(x)) - ^ ^ F e \ ,

(3.8)

F)^1)'S

- S(A + \b*{z) - ( y + F)^^L)

+ C'r-lC - SQS],

(3.9)

Important comment. At this place, we used the observability properties: the normal form (1.1) is crucial in the computation above. Now, we can make a time rescaling. We set: dt =

9(T)(1T,

e(r) = e(t),S(r)

oit=

6{v)dv, Jo = S(t),P(T) = P(t),e(r)

= 6(t),

267

to get the final set of equations: (i) | ( e ) = [(A - PC'r-lC)e («) Jt&

+ l(b(z) - b(x)) - %~$-Fe],

= [~(A + \b*{z) -{%

(3.10)

+ F)^=^)'5

- 5 ( ^ + \b*{z) - ( ^ + F ) ^ ^ )

+ C V ^ C - SQS],

{

" ° dt - X 0 ' First, there are some classical results allowing to bound the solutions of the Ricatti equation (3.10), (ii), for do > lj a n d A < 1. To apply these results, one has to notice that the linear time dependant systems: % = (A(u(t)) y=

+

\b*(,)-(^+F)X-^)x(t),

C(u(t))x(t),

are uniformly observable (in the sense of linear systems), for all bounded measurable functions ai(u(t)), b*j(z(t)),9(t), with o « > aj > am > 0. Precisely, we have: Lemma 3.2. If the functions a,i(u(t)), \b*j(z(t))\, 9(t), are all smaller than o-M > 0, and if a,i(u(t)) > am > 0, (which is the case by our assumptions), if 0 < A < 1, and 1 < 9(t) then, the solution of the Ricatti equation 3.10, (ii), satisfies the following inequality, a Id< S(t) < 0 Id, for all TQ > 0, for all t > To, where a and (3 depend on To, am,a,M (but do not depend on c, So > c Id !) This result is more or less classical. It is contained in 3 for instance. A detailed proof is given in 10 , because there are several mistakes in many textbooks. The key point for a simple proof is the precompactness of the weak-* topology on L°°[0, T], and the continuity of the input-state mapping of a control-affine system, for the weak-* topology on controls, and the uniform topology on trajectories x(t), t 6 [0, T]. Straightforward computations with (3.10) give: jt(e(t)'S(t)e(t))

< -Qm e'S(t)2e + 2e'S(t)(l(b(z) A ( l - 0 ) ,=,, ,

- b(x) - b*(z)e)) (3.11)

268

where Q > Qm Id. In particular, if t > T0, with a given by Lemma 3.2, this gives: jt(e(t)'S(t)e(t)) < HQma + ~ ^ - ) e'S(t)e +

(3.12)

2e'S(t)(i(b(z)-b(x)-b*(z)e))Using this equation, and again Lemma 3.2, we will now prove the theorem. 3.2.2. Proof of the short term estimation 3.2 This proof is in two steps. We will first prove an estimation for T > t > To > 0, and after for t < To. Gluing them together, we get the short term estimation 3.2. This is the standard high gain reasoning, and it is done in details in 10 for 6 constant. We omit the computational details. Step 1, T > t > T0. Straightforward computations using (3.12), Lemma 3.2 and the remark in Section 3.2 give: < e(T0)'5(To)e(To)e-(Qma-^))(t'To).

e(t)'S(t)e(t)

(3.13)

< /?||e(T 0 )|| 2 e~ ( c ? '" a _ ^ ; ) ) ( '" T o ) , and finally:

Therefore e(t)'S(t)e(t)

T > t > To :

(3.14)

\\e(t)\\2 < - e - ( Q - " Q - ^ ) ) ( t - r o ) | | e ( T o ) | | 2 . a Step 2, t < T0. We need a more straightforward estimation here. A very rough one is obtained just using Gronwall's identity. For certain s, k > 0, we have: ||P(i)ll<(l|P(0)||+*)e'T°.

(3.15)

We assume that 5(0) = 5o lies in the compact set: c Id < So < d Id. As a consequence, P(0) < -Id. By the equation (3.10), we have, for t < T0 : £(e) = (A- PC'r~1C)e + i(6(z) - b(x)) - m~^Fe, hence: ||e(t)H2 < l|e(0)|| 2 + f

l k - ( r ) | | 2 ( 2 P | | + 2||C|| 2 ||r- 1 || | | P | | + l)dr,

269 and by 3.15, we know that ||P(t)|| < 1, ||P(*)|| <
< ||e(0)|| 2 + P(T 0 ,c) f

\\e(r)\\2dr,

Jo

and P(To, c) is a positive decreasing function of c. Gronwall's inequality implies that:

|| £ -(t)|| 2 <*(r 0 ,c)|| £ -(o)|| 2 , with: \J>(T0,c) = epT°, ^(To,c) is also a decreasing function of c. In particular, ||e(T 0 )|| 2 < #(T 0 ,c)||e(0)|| 2 . Plugging this in (3.14), we get: HeWII2 < - e " ( 0 m a " ^ ) ( t _ T o ) * ( r 0 , c ) | | £ ( 0 ) | | 2 , for T > t > T0.

(3.16)

Hence, for T > t > T0, \\£(t)\\2 < - e _ ( Q m a " ^ ) t e ° " ' a T o * ( T 0 , c ) | | £ ( 0 ) | | 2 .

(3.17)

Going back to t < To, we have: l|e(*)H2 < *(To,c)|| £ -(0)|| 2 < *(ro,c)^|| £ -(0)|| 2 a Hence, in all cases (either t < To or To < t), we have: ||e(i)|| 2 < / / ( T o , c ) e _ ( Q m a " ^ > ) t | | e ( 0 ) | | 2 , 0 < t < T,

(3.18)

with H(T0, c) = f *(T 0 , c)eQmCcT°, a decreasing function of c. Therefore, going back to the initial time r, since t = J^ 6(v)dv, and t < T, then, r < T(T), and t >

6(T(T))T:

\\e(r)\\2 < F(To,c)e-(°'" a e ^( T »-' L ') r ||e(0)|| 2 ,T(T) > r > 0, if C = Qma6(T{T))

- L' > 0, which is implied by 6o>eXT{-T\-^-\)

indeed, if (3.19) holds, since

0{T{T))

+ l,

(3.19)

= 6{T) = 1 + (0O - l) e - A T < T ) > ^

.

270

Since e

=

A^i,

1, ||e(r)|| 2

and 9 >

<

IKA-1)!!2^)!!2

<

^ ( " - ^ H ^ T ) ) ! 2 , we get, for all r 0 > T > 0 :

||e(r)|| 2 < ^ ( " - ^ ( ^ ^ ( T c ^ e - W - ^ ^ - ^ ^ H e ^ ) ! ! 2 , fore0>eAro(?^

1) + 1,

jj

or equivalently, 6{TQ) > —

.

H(T0, c) is a decreasing function of c. This is the short term estimation (3.2). If A = 0, it gives the standard high gain estimation. 3.2.3. Proof of the long term estimation Going back to (3.12), and using Lemma 5.2, in Section 5, we get, for all A, 0 < A < 1, t > T0, < - * i e'S(t)e + k2 9(t){n-2)\\S\\

jt(e(t)'S(t)e(t))

\\e\\\

where k\ = Qma, k2 is a positive constant. Lemma 3.2, applied to the Riccati equation in (3.10), implies: jt(e{t)'S{t)e{t))

< -h

e'S{t)e + k'2 #"- 2 > \\e(t)'S{t)e{t)\fi,

(3.20)

for another positive constant k'2. Now, we apply Lemma 5.1, in Section 5, to get that, for t > T > To: e(t)'S(t)e(t)

< 4e-klit-T^e(T)'S(T)e(T),

(3.21)

as soon as m

e(T)'S{T)e{T)6{T)^-V

Setting, q = e(T)'S(T)e(T)6(T)2(-n-V, tion (3.18). It gives q < /?F(T o ,c)e~ q<

(Qma

<

\2

^ - .

let us use the short term estima~^ ) T ||e(O)|| 2 0(T) 2 ("- 2 >,

PH(To,c)e-{Q™a-&)T\\e(0)\\%{n-2)-

If: 0o > eXT(-

IV

1) + 1,

(3.22)

271

then 2 a a _ _L_ > 22L^1 . indeed, in that case, 6{T) > 6(T) = 1 + (80 Then, let us chose T = T* - Log( ffi,"1 )* > T 0 (in order to get the equality in (3.22)). This is possible, since we can assume from the very beginning that J ^ - 1 > 0 (we can increase V for this) and ffir1 - > eT° > eXT° (we can take 0O large enough). iV

_ j

< ^(r 0 , c )(fe- r )^|!e-(o)||X 2(ri - 2)

q

< /m(To,c)|| e -(0)|| 2 (2(-^ - l ) )

2

^^-

2

)-

9

^.

Then, if:

(3 23)

"<W^Y

-

for 0O large enough, for ||e 0 || bounded, q is arbitrarily small. This means that the property (?#) above is met at T = T*(00, A), as soon as A satisfies (3.23) and 8Q is large enough. In that case, (3.21) above holds, for t > T* (> T0) : e(t)'S(t)e{t)

< 4e-* 1 ('- T *>e(T*)'5(:r)e(T*), <

4e-klt(^j2^-)^e(T*yS(T*)e(T*).

This implies, with (3.18): '|e(t)H2 <

de-^(^~L)^\\e(T*^ \ $ HIT

„\„L'T*-k!t/

#0-1

<4^H(T0,c)e^'e-^(^^-)^\\s0\\2, < A^-H{T0,c)e-^{^

2L'

ti

-)>\\e0

Qma

for t > T* (> T 0 ). For £ < T*, using (3.18), and the fact that k\ = Qma : || £ -(i)|| 2 < J ff(To, C )e-^ t e L ''|| e o || 2 ,
272

because /fl

1

> 1.

Therefore, for alU > 0 :

\m\f
-

ff(To>c,A)e-*'i0oil*£:||eo||2, e~Xr),

Finally, IKr)||2 < H(To,c,A)e-fc"|ko||2«'oil^+2(n-1)e-*1*(1-e-*T), where H(To,c, A) is a decreasing function of c. This is the long term estimation. It holds as soon as A satisfies (3.23), and for 6Q large, depending on A. 3.3

Practical implementation:

making the observer "recursive"

We consider a one parameter family {OT,T > 0} of observers of type (3.1), indexed by the time, each of them starting from So, 9g, at the current time T. In fact, in practice, it will be sufficient to consider, at time r, a slipping window of time, [r — T, r[, and a finite set of observers {Ot{, T -T
273

OtN- Then, the error will be given by the "long term" and "short term" estimates at time T : \\e(T + \\E(T + T)\\2 <

T)\\2
a. If T is large enough, the asymptotic behavior will be the one of the "extended Kalman filter". b. At the beginning, the transient is the one of the HGEKF. c. the error can be made arbitrarily small in arbitrary short time, provided that 6Q is large enough. 2. If at a certain time we have a "jump" of the state, then, the innovation of the "old observers" will become large. The "youngest" one will be chosen, and the transient will be the same as the one of the HGEKF, first, and of the EKF, after T. This looks very promising. We show on an example in the next section, that it works very well. 4 4-1

Application: observation of a binary distillation column The constant molar overflow model

The model we consider is the classical "constant molar overflow" (CMO) model. It is one of the most simple distillation models, and it is used by many process engineers (for instance, even in its static form, it is used for simple short-cut distillation calculations). Since everything here follows from the very special " tridiagonal" structure of this model, and since any reasonable distillation model possesses such a structure, all what we do in this paper can certainly be extended to more precise distillation models. The equations are based upon: a. a thermodynamical relation describing the thermal equilibria for each tray. b. Material balances on each plate. Thermal balance on each plate is replaced by the "Lewis hypotheses", that lead to the fact that the liquid and vapor flowrates along the column are constant in the "stripping" (above the feed) and "rectification" (below the feed) zones. For justification of these assumptions, see 1 5 .

274

The equations of this model are: Total condenser: dx\ Hi^j-= (V + FV)(y2 -

Xl).

(4.1)

Rectifying section, j = 2, • • • , / — 1 : dx'

(4.2)

Feed tray: Hf^-=

FL(ZF + L(xf-!

Xf)

+ FV(k(ZF)

- Xf) + V(yf+1

-

- yf)

(4.3)

yf).

Stripping section, j = f + 1, • • • , n — 1 : Hj-±

Vj).

(4.4)

Bottom of the column: dx Hn-jr = (L + F I ) ( i „ _ i - xn) + V{xn - yn). at The parameters have the following physical meaning:

(4.5)

n

f Hi Xj

Vi FL,FV,L,V ZF

= (L +

FL){XJ-!

-

XJ)

+ V(yj+1

-

number of trays, index of the feed tray, liquid hold up on the j t h tray (a geometric constant), liquid composition on the j t h tray, vapor composition on the j t h tray, feed (liquid and vapor), reflux and vapor flow, feed composition (molar fraction of light component in feed).

On each tray the liquid and vapor compositions, Xj and yj, are linked by the liquid/vapor equilibrium law yj = k(xj). We assume that the function k is monotonic, i.e. we do not consider azeotropic distillation. Each of the equations is relative to a tray. It just expresses the accumulation of the liquid on the corresponding tray, and the thermodynamical equilibrium. The condenser and the bottom of the column are assimilated to tray 1 and tray n respectively. The state of the model is the liquid composition profile of the more volatile component on each tray, denoted by (XJ).

275

The top and bottom product compositions x\ and the two observed variables. In practice, they are also the two variables that one wants to control: they are the "qualities" of the products going out of the column. The two control variables are the reflux flow-rate L and the vapor flowrate V. There are also two disturbances to be counteracted: a. changes in the feed rate F = FL + FV. In general this is a "measured disturbance", (a flowrate measurement), b. the in-feed composition Zp. In general, it is unknown, and it is practically very expensive to "observe it". Moreover, it may change brutally. We will consider this feed composition Zp as an unknown (constant) state variable. When Zp changes, the consequence is a jump of the state of the system. The qualitative properties of this model are very nice (see 10 , 18 , 1 7 ) : a. For positive control variables L and V, (negative doesn't physically makes sense), the "physical" domain D = [0,1]" is positively invariant under the dynamics. This means that all the state variables Xj remain between 0 and 1. b. In the domain D, all other variables (than the ar^'s and the t/j's) being constant, there is a single equilibrium, which is globally asymptotically stable. c. It has very nice observability properties, as will be discussed later on. Our goal in this section is to construct an estimator of the state x, and more specifically of the feed composition Zp, by using the results of the previous sections. 4-2

Observability of the model and synthesis of the observer

A complete analysis of observability and observer synthesis has been carried out in 10 in the general case. It happens that, even if the feed is considered as an unknown state variable (meeting the equation ^~f- = 0), the model is observable in the strongest possible sense. In particular, as we shall see, it can be put in a normal form similar to (1.1). Our purpose here is just to apply the observer described in the previous sections. Hence, we will fix a special case of distillation column. But all what we show works in general. We will chose:

276 • n = 5 and / = 3, • The function k is a diffeomorphism from [0,1] into itself and is given by, ax

ur \ k(X) v

=

;

T7—.

' 1 + (a - 1) x Here a is the "relative volatility" of the mixture. It is a physical parameter larger than 1 (but close to 1). The closer to one, the most difficult distillation. If a = 1, the two products are thermodynamically identical, and cannot be distillated (the model is not controllable). • Let us observe that k is a diffeomorphism from

—J^I,+OO

to

]-°°.=£r[• The feed is assumed to enter the column at its "bubble point". As a consequence, F = FL. Let us make the following change of state variables: £i = x\, £2 = k (x 2 ), £3 = x3, £4 = i 4 , & = x5 and £6 = ZF. Then, the system can be rewritten as:

H2$- = k' {k-1 ( 6 ) ) (L(& - fc-1 (&)) + v(k ( 6 ) - & ) ) , , # 3 # = Ffa - 6 ) + L(k~l ( 6 ) - 6 ) + V(k (&) - k ( 6 ) ) , S ff4% = (L + F ) ( 6 - £4) + V(k (&) - A (&)),

(4-6)

tf5ff = (L + F)(& - &) + V(& - * (&)), #B% rf< = 0

or: d6 = A(L,V)Zt dt

+

b(L,V,tt),

where,

A(L,V)

0 0 = 0 0 \0

0 0 0 0 0

0 0 L+F

0 0

0 0

0 0 \ 0 0

0 0 0 ^ 0 0 0 0 0/

(4.7)

277

and,

I

-—£ 1

\

1

V (AT fa)) (Lfa - fc- fa)) + V(k fa) - &)) / t f 2

(-F& + ^(fc-1 (6) - 6) + V(kfa)- k fa))) /tf 3

b(L,V,0 =

(-(L + F)£4 + V{k fa) - A fa))) / # 4 (-(£ + F)& + V(& - A fa)))/H5 0

\ /

6i(V,6)

/

\

b2(L,V, £!,..., e5) h(L,V, &,&,&) h{L,V,(ji) 0 \ The observations are then given by V =

J

100000 000010

f = Cf.

Now, since in fact the only pertinent (and positively invariant) part of the state space is D' = [0, l ] 6 , we can manage the things for b be compactly supported, as in section 1.1, and unchanged on D'. Let us change b (L, V, £) in the following way outside [0,1] : replace b (L, V, £) by b(L,V,0 = b(L,V,*fa) where $ fa,. . . , & ) = (if fa),... ,
^ , —^r • This modification does not change the "physical

J

a —

2

a

2 I

trajectories". Our system has the property to be observable for any input, as soon as the control variables L and V are > 0. Here, we assume that L, V are bounded from below (and from above) by > 0 constants: LM > L(t) > ei > 0, VM> V(t) > e 2 > 0. This assumption is the analog of the assumption 0 < am < ai(u) < aM, in section 1.1. It is a realistic requirement from the physical point of view. To finish, let us point out the fact that we are in case 1 of section 2.1 above (i.e. the nongeneric case): The number of observations is equal to the number of control variables (it is 2).

278

Due to these observability properties, we will be able to apply the observer of the previous section 3.3. In fact, it will be an adaptation of the results of section 3, Theorem 3.1, to this multi-output case. We leave the reader to check (this is really straightforward) that all the reasoning in the proof of Theorem 3.1 can be strictly repeated, and that the statements of this theorem are valid for the distillation column. Of course, in practice, we didn't compute the theoretical bounds Ao and #o(A). We have just got some values for them by experimentation. Also, the number TV of "parallel" observers, and the "sampling times" U of section 3.3 have been chosen experimentally. Finally, the state of our observer is the collection of the states of TV independent observers (zi, 5j, #i) i = 1 N. Each observer is a set of three equations of the following form: f =A(u)z + b(u,z)-S (ty1 CTR-6X [Cz - y (t)) H = - (A(«) + b* (z,«))'S-S(A(u) + b* (z,«)) + C'R^C

§=A(l-0)

-

SQ9S

where u = (L, V). Due to the multi-output structure, with "Brunovsky-like" blocks of different dimensions (4 and 2), a way to make the proof of Theorem 3.1 work, is to take a matrix R depending also on 6, as shown below. This could be avoided by increasing the dimension of the state as explained in 10 . It is not hard to check that a good choice is to set: A = d i a g ^ — , ^ , — ,g,l, with Qe = 92A~1QA~1

and Re = (CA^C)

R

p J (CA^C).

In practice, we have chosen TV — 5 observers, and we have taken a regular sampling ^ . That is to say, at each time step k^, the oldest observer is replaced by a new one (with 6 = #o and a new guess of state and covariance matrix). At the beginning of the simulation, we chose an initial value #o of 6 for each observer, such that the ith observer has #; = 1 + e~ A — ^ (0O - 1), see figure 3, where "crosses" represent reinitializations. We have implemented our observer as described in the previous section. Since the state has dimension 6, each observer requires to solve 28 ordinary

279

differential equations (for the state, the Riccati matrix, and the very simple equation for 6). Finally, our observer is a set of 140 ODE's. We have solved it in conjunction with the model (6 equations) using LSODAR from ODEPACK ( 14 ), without taking into account the possibility of decoupling these equations (which are indeed equivalent to five systems of 34 equations, including the model into each system). A simulation of 3 hours of real time takes about 40 seconds on a Pentium III machine.

4-3

Simulation results

We have chosen the following constant parameters: • Hold-up Hi = 40, Hj = 10 for j = 2,3,4 and H5 = 80, • Relative volatility a = 2. We have applied the following scenario: - During the simulation, the state noise is simulated by the sum of several sine functions at some random frequencies representing a band limited noise with an amplitude of 1 0 - 8 before the time *2 = 116 mn 40 s and 10~ 2 after this time, - Moreover, at time t\ = 66 mn 40 s, we simulate a step in the feed quality Zp from 0.45 to 0.60. Hence we can consider that there is no perturbation before time t\, where a large "jump of the state" occurs, - after that, nothing happens until time t 4.3 which is running.

280

Finally, R is equal to 10 2 times the 2 x 2-identity matrix and Q is 10 times the 6 x 6-identity matrix.

9

First of all, the behaviour of the observer is very good during the unmodelled transient as well as during smooth operation, see Figure 4.3: top and bottom quality measurements are plotted, as well as the unknown feed quality, each curve being represented by a continuous line. The overall estimation of the feed quality, corresponding to the estimation of the feed quality provided by the observer with the smallest innovation, is represented by a dashed line. It is very close to the actual feed quality.

20

40

60

80

100

120

140

160 180

Figure 1. Measured output and estimation of the feed quality.

A more accurate plot is presented on Figure 2 where we have only shown the relative estimation error of the feed quality. The estimation provided by the best observer (in our sense, that is to say, the observer with minimal innovation) is the continuous line. The crosses represent the estimation of Zp provided by other observers every minute. One can see that our criteria on the innovation to select the right observer is a good choice, at least in this case. Moreover, the behavior of the observer is very close to what we expected from the theoretical results:

281

Figure 2. Relative error between the actual feed quality and its estimation by the selected observer (continuous line) and the others.

- When no perturbation arises, the best observer (that is to say the observer with the smallest innovation) is the one with the smallest value of 6 i.e. the oldest observer which is also the observer which is the closest to the pure extended Kalman observer. -If a large perturbation occurs (such as the feed change at time t\ = 66mn40s), the best observer becomes the youngest one, i.e. the observer with the highest 9. -Of course, small perturbations are well corrected by oldest or intermediate observers. This is very clear on the figure 4. Our conclusion, from these simulations, is that even if the use of several observers in parallel requires the introduction of new tuning parameters (80, A, N and T), the choice of these new parameters is very easy, due to their very clear effect on the results. ^From a practical point of view, 60, A, N and T have to be chosen such that at any time, there is an HGEKF and an EKF-like observer running at the same time, that is to say such that 1 + e~x^ (6Q — 1) is large enough (to ensure that at least one observer is a HGEKF) and such that l + e _ A T (60 - 1) is close to 1.

282

0

20

40

60

80

100

120

140

160

180

Figure 3. T h e 5 observers. Time of reinitialization of each observer ( x ) , and the best one (continuous line).

0

20

40

60

80

100

120

140

160

180

Figure 4. Various values of 6 versus time (dotted lines), and best observer (continuous line).

283

Also, an important point, for people that are used to tune Kalman's observers, is that the choice of the Q and R matrices is less crucial than with a single observer which has to be tuned in order to be efficient both with and without perturbations. Moreover, this approach allows us to obtain a diagnosis of abnormal behavior: if the smallest innovation is provided by the last reinitialized observer then one can conclude that the model has encountered a perturbation. If this happen for a long time then one can conclude that the model has some difficulties to deal with certain unmodelled perturbations. Indeed, the scenario that we have applied in our simulations can be easily deduced from the figure 4. 5

Appendix. Technical lemmas

Lemma 5.1. Let {x(t) > 0, t > 0} C R n be absolutely continuous, and satisfying: — < —Xx + kxi/x, at for almost all t > 0, for X,k > 0. Then, as soon as x(0) < jjp, x(t) < 4x(0)e-'\ Proof. We make the successive following changes of variables: y = y/x, z = 1/y, w(t) = e~2(z(t). Then, all the quantities y(t), z(t), w(t) are positive and absolutely continuous, on any finite time interval [0,T]. We denote by ' the derivatives with respect to time. Straightforward computations give, for almost all t > 0 :

y'<-\y ,

A - 2

+ \y\ k 2

w' > - e ~ ^ ' - . 2 Moreover, w(0) =

* . Then, for all t > 0,

(5.1)

284

If

}

— j > 0, then w(t) > 0,and we can go backwards in the previous

inequalities: - -(1 - e - ^ )

w(t) > - .

y(t) <

x(t) <

**'{-&* -i)+r /x(0) x(0)e~xt (1-V^(0)!)2'

Hence, if z(0) < ^ , or 1 - y/xJU)^ > §, then: x(t)

<4x(0)e~xt. D

Lemma 5.2. Let B — b{z) — b{x) — b*(z)e be as in Section 3: e = z—x, b(x) = A6(A _1 a;), b*(z) = A6*(A _ 1 x)A _ 1 , where b*(x) is the Jacobian matrix of b at x, and where b is compactly supported. A = diag{\, |,—, fphrr), 9 > 1. Then, \\B\\ < K <9"-111£:||2, for some K > 0. Proof. Let us consider a smooth expression E(z, x) of the form: E(z,x)

= f(z) — f(x) — df(z)e, with e = z — x,

where / : W —> R is compactly supported. We have, for t > 0: f(z - te) = f(z) - »

/ |£(z -

re)dr,

and: 3/ 9a;

>-"> = £«-£•> jf sis*-*)*

Hence, ,1

f(z-e)

=/

M-S>£«

rr

a 2 /

+ ! > * / [j!&f-«*»*•

285

Since / is compactly supported, we get: M p \f(z) - f(z - e) - df(z)e\ < y £

l^l>

whereM = s u P J 5 ^ - ( x ) | . Now, we take f = bk, and we use the facts that 6* depends only on xi, •••,Xk, and t h a t 0 > 1 :

'dxidxj ^ ' W I ^ ' - ' l ^dxidxj lA-'x),. This gives the result.

•

References 1. M. Balde, P. Jouan , Observability of control affine systems, ESAIM/COCV, 3, 345-359 (1998). 2. J.S. Baras, A. Bensoussan, M.R. James, Dynamic observers as asymptotic limits of recursive filters: special cases, SIAM J. Appl. Math., 48, 11471158 (1988). 3. R. Bucy, P. Joseph, Filtering for stochastic processes with applications to guidance (Chelsea publishing company, 1968, second edition, 1987). 4. F. Deza, Contribution to the synthesis of exponential observers (Ph.D. thesis, INSA de Rouen, France, June 1991). 5. F.Deza, E.Busvelle, J.P.Gauthier, High-gain estimation for nonlinear systems, Systems and Control Letters 18, 295-299 (1992). 6. J.P. Gauthier, H. Hammouri, I. Kupka, Observers for nonlinear systems; IEEE CDC Conference, Brighton, England, december, 1483-1489 (1991). 7. J.P Gauthier, H. Hammour i S. Othman, A simple observer for nonlinear systems. IEEE Trans. Aut. Control, 37, 875-880 (1992). 8. J.P. Gauthier , I. Kupka, Observability and observers for nonlinear systems. SIAM Journal on Control, 32, 4, 975-994 (1994). 9. J.P. Gauthier , I. Kupka, Observability for systems with more outputs than inputs. Mathematische Zeitschrift, 223, 47-78 (1996). 10. J.P. Gauthier , I. Kupka, Deterministic observation theory and applications, (book to appear at Cambridge University Press). 11. A. Jaswinsky, Stochastic processes and filtering theory, (Academic Press, New York, 1970). 12. P. Jouan, Singularites des systemes non lineaires, observabilite et observateurs (Ph.D. thesis, Universite de Rouen, 1995).

286

13. P. Jouan, J.P. Gauthier, Finite singularities of nonlinear systems. Output stabilization, observability and observers, Journal of Dynamical and Control Systems, 2, 2, 255-288 (1996). 14. A. C. Hindmarsch, Odepack: a systematized collection of ode solvers, in Scientific Computing, R.S. Stepleman et al. (Eds.),55-64, (NorthHolland, Amsterdam, 1983). 15. C D . Holland, Multicomponent Distillation, (Englewood Cliffs, NewJersey, USA: Prentice Hall, 1963). 16. J. Picard, Efficiency of the extended Kalman filter for nonlinear systems with small noise, SIAM J. Appl. Math., 5 1 , 3, 843-885 (1991). 17. H.H. Rosenbrock, A Lyapunov function with applications to some nonlinear physical systems, Automatica, 1, 31-53, (1962). 18. P. Rouchon, Simulation dynamique et commande non lineaire des colonnes a distiller (These de l'ecole des mines de Paris, 1990). 19. F. Viel, Stabilite des systemes non lineaires controles par retour d'etat estime. Application aux reacteurs de polymerisation et aux colonnes a distiller (These de l'universite de Rouen, 1994). 20. F. Viel, E. Busvelle, J.P. Gauthier, A stable control structure for binary distillation columns, International Journal on Control, 67, 4, 475-505 (1997).

287

LIE SYSTEMS I N CONTROL THEORY JOSE F. CARINENA AND ARTURO RAMOS Departamento de Fisica Teorica. Facultad de Ciencias, Universidad de Zaragoza, 50009, Zaragoza, Spain E-mail: [email protected], [email protected] We show that the theory of systems of differential equations allowing a rule for expressing the general solution in terms of some particular ones, the so-called Lie systems, are relevant in Control Theory. These systems are related to some particular cases of control systems on Lie groups, whose exact solution is usually found by means of a method generalizing the one proposed by Wei and Norman. We establish some results concerning these systems. The general theory is illustrated with some examples.

Dedicated to Professor Velimir Jurdjevic on the occasion of his 60"" birthday.

1

Introduction

There exist systems of first order ordinary differential equations such that the general solution can be expressed in terms of a number of fundamental solutions and several constants determining each particular solution. The characterization of such systems is due to Lie 19 and henceforth will be called Lie systems. They are receiving a lot of attention during these last years, because they appear very often in a variety of problems in mathematics, physics and other branches of science and engineering 6 2S ' . The two best known examples are the linear systems and the scalar Riccati equations, but Lie systems also appear when studying evolution in Lie groups. More specifically, as far as Control Theory is concerned, drift-free right-invariant affine control systems in Lie groups 3 ' 1 6 are another example of Lie systems. The important fact we want to stress here is that Lie systems in Lie groups are in some sense prototypical, because any other Lie system can be related with one of such models. The aim of this paper is therefore twofold: to expose how Lie systems are interrelated with systems on Lie groups, and how a generalization of the method proposed by Wei and Norman 23 ' 24 can be useful for solving this last problem. There also exists reduction techniques for these problems 7 which will not be considered here. We feel that a careful study of Lie systems from a geometric viewpoint may be useful for a better understanding and treatment of these systems, which appear quite often in important problems of geometric Control Theory. The paper is organized as follows. Next section is devoted to the description of Lie systems and the formulation of the Theorem which characterize them, along with some simple but important examples. Then, we relate each of these systems with a similar system but denned in an associated Lie group. We follow in Section 3 with some questions of interest in Control Theory concerning these systems. In

288 Section 4 we show how a generalization of the Wei-Norman method applies in these problems. Finally, we illustrate in Section 5 the theory developed by means of several examples. 2

Systems of differential equations admitting a superposition rule

An interesting result by Lie is the characterization of time-dependent systems of first order differential equations ^M=Xi{x,t),

i = l,...,n,

(2.1)

for which it is possible to write the general solution in terms of a set of m arbitrary but independent particular solutions { x ( i ) , . . . , £( m )} by means of a function also depending on a set of n constants, X(j) = F(x(i),... , X ( m ) , c i , . . . ,c„). The function F will be called a superposition function. From the geometric viewpoint the system (2.1) can be considered as the one determining the integral curves of the timedependent vector field n

X(x,t) = J2xi(x,t)^-.

(2.2)

i—l

For more details about the geometry of these objects see, for example, 5 ' 6 . The characterization by Lie 1 9 of such systems, hereafter called Lie systems, amounts to say that they correspond to time-dependent vector fields which can be written as a linear combination r

X(x,t) = Y,bct(t)YM(x) a= l

with time-dependent coefficients ba(t) of vector fields Y^a\x) closing on a real rdimensional Lie algebra, i.e. { Y ^ } are linearly independent and there are r 3 real numbers, cal3 7 , such that r

[ y ( a ) , y , w ] = J2 c Q " 7 ^ ( 7 ) •

(2.3)

7=1

Moreover, the number r satisfies r < mn. For a geometric proof of this Theorem, see 6 . Simple examples of Lie systems are the linear systems ^ - = £ " L i A* j(t)x3, whose superposition function is linear, F(x(i-), . . . , X(ny,k\, . . . , k„) =fciX(i)+ - • • + A;„X(n), and the scalar Riccati equation x = /2(i) x2 + fi(t) x + fo(t), for which the superposition function reads as Xi(x3-x2) F(xi, x2, x3; k) = —

+

.

kx2(xi-X3) r— .

(X3 - X2) + k(Xl - X3)

289 See 7 ' 8 for details. Another remarkable example occurs when one takes a Lie group G and considers vector fields Xa in G which are either left-invariant or right-invariant, corresponding to the Lie algebra g of G or to the opposite algebra, respectively. We refer to standard references as 9 ' 1 7 for the properties of Lie groups and of their actions on manifolds. Let us choose a basis { a i , . . . , a r } for the tangent space TeG at the neutral element e 6 G. If Xa denotes the right-invariant vector field in G such that X
ff(t) = - £ > « ( * ) * « OK*))-

(2-4)

a=l

Applying (-R9(t)-i )»9(t) to both sides of this equation we obtain the equation on TeG r

(J*,(t)-i).,(«)($(*)) = " E M « K ,

(2.5)

a= l

which we will write with a slight abuse of notation as (S3 _ 1 )(t) = — 5Za=i ba(t)aa, although only in the case of matrix groups .R 9 (t)-i» 9 (t)3W becomes the matrix product g(t) <j(t) _1 . Analogous expressions are obtained if left-invariant vector fields Xt in G are considered, defined by X ' ( e ) = o 0 , that is, g(t) = ^ = 1 ba{t)X%(g(t)) and (i s (i)-i).j(()(j(*)) = E U i ba(t)aa. Now, let H be a closed subgroup of G and consider the homogeneous space M = G/H. Then, G can be seen as the total space of a principal bundle (G, ir, G/H) over G/H, where 7r : G —> G/H is the canonical projection n : g \-¥ gH. It is known 6 T ' that the right-invariant vector fields Xa are 7r-projectable and that the 7r-related vector fields in M are the fundamental vector fields —Xa = —Xaa corresponding to the natural left action $ of G on M, i.e. n*g(Xa(g)) = —Xa(gH). In this way we have an associated Lie system on M corresponding to the time-dependent vector field r

X{x,t)

= ^ba{t)Xa{x),

(2.6)

a=l

where we have taken x = gH. Therefore, see 6 ' 7 , the solution of (2.6) starting from xo is x(t) = $(g(t),xo), being g(t) the solution of (2.5) with g(0) = e. In other words, the curves x(t) so defined satisfy T

x(t) = Y,ba{t)Xa{x{t))

(2.7)

a= l

and the given initial condition. Conversely, given a system of type (2.6) such that Xa be complete vector fields and close on a Lie algebra g, we can regard them as

290 fundamental vector fields relative to an action defined by the composition of the flows of the vector fields Xa. Then, the restriction to an orbit will provide an homogeneous space of the type described above. As a consequence, we can reduce the problem of finding the general solution of the system giving the integral curves of (2.6) to the one of finding one solution of an equation of type (2.4) on the group G. This may give rise to more complicated equations than the original ones. However, there are two main advantages in doing this. Firstly, because of the right-invariance of (2.4), one only needs to find the particular solution starting from the neutral element, g(0) = e. Secondly, the solutions for all Lie systems corresponding to all possible (effective) actions of the same group G on the same or different manifolds, are obtained at once from the solution in the group.

3

Control and controllability of systems on Lie groups

Equations of type (2.6), where the functions ba(t) play the role of the control functions, appear often in Control Theory. These functions are to be determined in such a way that, for example, the trajectory passes through one or two specific points in the configuration space, or maybe gives some cost functional a stationary value. If the manifold in which the vector fields are defined is a Lie group the equation will be r

*(«) = £ &«(*)*-(*(*))•

(3-1)

Q= l

When the vector fields Xa in the Lie group G are right-invariant (respectively leftinvariant) the control system is said to be right-invariant (resp. left-invariant). As an example, the system (2.4) can be regarded as a right-invariant control system. The important question of controllability of control systems on Lie groups like (3.1) has been analyzed by Brockett 3 and Jurdjevic and Sussmann 16 . They have shown that controllability of such systems can be determined by studying algebraic properties of the corresponding Lie algebra g. We offer next a simple proof of one of their results. T h e o r e m 3.1. A drift-free right invariant system of the form (3.1) on a connected Lie group G is controllable if and only if the Lie algebra generated by {Xi, • .. ,Xr] is 0Proof. If h is the Lie algebra generated by { X i , . . . ,Xr}, then the Lie algebra of the Lie system is not g but the subalgebra f). The orbit of the neutral element e € G is the subgroup H of G with Lie subalgebra f). It is then clear that if h is a proper subalgebra of G, the system is not controllable, while it is so when h = g. • Note also that affine control systems on a Lie group like T

g(t) = X0(g(t)) + J2 ba(t)Xa(9(t)), a= l

(3.2)

291

with {Xo, Xi, . . . , X,.} right- or left-invariant vector fields, can be obtained (locally) from

^• = J2da(r)Xa(h(r))

(3.3)

Q=0

by means of a reparametrization given by dt/dr = do(r), ba(t) = da(T(t))/do(r(t)) and g(t) = h(r(t)). Therefore, the result is applicable for systems like (3.2) such that {Xo, X\, ..., Xr} close on a Lie algebra.

4

The Wei—Norman method

The equation in the group (2.4), or more precisely, in TeG given by (2.5), can be solved using a generalization of the method proposed by Wei and Norman 2 3 ' 2 4 for linear systems of type dU(t)/dt = H(t)U(t), with U(0) — I. It is based on the following property 7 : If g(t), gi(t) and gi{i) are differentiable curves in G such that g(t) = gi(t)g2(t),Vt <=R, then,

= ii M (0-i. 9 1 («)(9i(*)) + Ad(»i(0){iZ w (t)-i. M ( o(S2(*))} • (4-1) The generalization of this property to several factors is as follows. Let now g(t) be a curve in G given by a product of other I curves g(t) = gi{t)g2(t) • • • gi(t) = Ili=i 9i(t)- Then, denoting hs(t) = I l L . + i 9i(t) for s e {1, . . . , / - 1} and applying (4.1) we have Rgw-i

. 9( «)(fl(0) = R9lW-i

.„(«)(si(0) + A d ( S i ( t ) ) { i i k l ( t ) - i . h l ( t ) ( f t i ( t ) ) } .

Since hs(t) = ga+i(t)ha+i(t) for s 6 {1, . . . , I — 2}, the process can be iterated, and using that Ad(gg') = Ad(g) Ad(#') for all g, g' 6 G we obtain

« 9 (t)-i .,(«)($(*)) = E

A d

( n * - ( * ) ) {R9iM-*

*9iit)(9i(t))}

= E ( I I Ad (*(*)) ) {*«(*>-! .«(t)(9i(*))} .

(4-2)

where it has been taken go(t) = e for all t. The generalization of the Wei-Norman method consists on writing the curve g(t), solution of (2.5), in terms of the second kind canonical coordinates w.r.t. a basis of the Lie algebra g, {m, ..., ar}, for all t, g(t) = YYa=1 e x p ( - u a ( t ) a a ) , and transforming the differential equation (2.5) into a differential equation system for the va(t), with initial conditions va(0) = 0 for all a = 1, . . . , r. The minus signs in the exponentials have been introduced for computational convenience. Then, we use

292 the result (4.2), taking I = r — dimG and ga(t) = exp(—v a (t)a a ) for all a. A simple calculation shows that Rga^-i „ 9Q ( ( )(pa(*)) = — va(t)aa, so that (4.2) reduces to R

g(t)-i *9(t)(g(t)) = - ^ Z * " I I I Ad(exp(-«^(t)o^)) J aa

= ~ ^2 Vac I Yl exp(-V0(t)

ad(a0)) J aa ,

a=l \0
a

I I I exP(-v0(t)a.d(a0)) \0
J aa = ^ 6 a ( t ) a a , J a=l

(4.3)

with va(0) = 0, a = 1, . . . , r. The resulting differential equation system for the functions va(t) is integrable by quadratures if the Lie algebra is solvable 2 3 , 2 4 , and in particular, for nilpotent Lie algebras. 5 5.1

Illustrative examples Model of a mixed-culture

biorreactor

Our first example is inspired in the Example 2.44, p. 44 of 22 , related with the model of a mixed-culture biorreactor. The state space will be lRn, where we take coordinates (xi, . . . , x„). The control system of interest is ii — bi (t)xi

if

i/ j ,

±j = 6i (t)xj + b2,

(5.1)

with j 6 {1, . . . , n} arbitrary but fixed. Let us consider the vector fields

dxi '

dx

The control system (5.1) therefore determines the integral curves of the timedependent vector field b\(t)Xi(x) + b2{t)X2(x). Taking the Lie bracket of X\, X2, 71

[Xi,X2]

Q

Xi

^ dxi'dxj

we see that {Xi,X2} close on a Lie algebra isomorphic to the Lie algebra of the affine group in one dimension Ai. The elements of this group can be represented by a pair of real numbers (ri, r2) with r\ > 0, with the product law ( n , r2)(r\, r2) =

293 ( n r ' i , r\r2 + r2). The corresponding Lie algebra has a basis {ai, a2} in which [ai, 02] = — <X2. The adjoint representation of this Lie algebra in this basis takes the form ad(a1)=(°_°1),

ad(aa)=(5J)

and therefore exp(-wi ad(aO) = ( 0 e «i J >

exp(—u2 ad(a 2 )) = [ _v

l

)•

If we express the solution of the associated problem in the group A\ as the product g(t) = exp(—vi(t)ai)exp(—V2{t)a,2), then (4.3) reduces in this case to vi ai + i ) 2 e x p ( - w i ( i ) a d ( o i ) ) a 2 = viai + v2 e" l ( t ) a 2 = 6i(t)ai + b2(t)a2 , and then, i2=e-Vl(t)b2(t),

vi=bi(t),

jointly with «i(0) = v2(0) = 0. The integration is immediate, giving vi(t) = b J*bi(s)da and v2(t) = J* e~ K ^'^ds'b2(s)ds. Going back to the initial manifold, we know that X\, X2 are the generators of dilations in R n and translations in the j direction, respectively, so the action can be written as $ ( ( M I , U2), (XI, ...,

Xj, . . . , xn))

= ( e u i x\, . . . , e" 1 (XJ +

1

u2), . . . , e" xn), where the group has been parametrized now by the second class canonical coordinates, in the same factorization order as the one chosen for the solution curve in the group, i.e. ui = l o g n , u2 = r2, so that the group law reads as ( « i , u2)(u[, u2) = (u\ + u\, eui u2 + u2). Then, the general solution in R" is $((vi(t),v2(t)),(xio,...,xjo,...,xno)) = ( e B l ( t ) xio, . . . , e B l ( t ) (xj0 + Bl< 3) Bl(4) f*e~ - b2(s)ds), ..., e x „ 0 ) , where Bi(t) = f*bi(s)ds. The result is exactly the same as the one obtained by direct integration of the system (5.1). 5.2

Model of maneouvring

an automobile

or of a robot

unicycle

22

This example is inspired in the Example 2.35, p. 52 of , related with a simplified model of maneouvring an automobile. Essentially the same control system arises as well in the problem of path planning for a robot unicycle, see e.g. 18 and Eq. (4) of 2 0 . The configuration space is R 2 x S 1 , where we take coordinates (xi, x2, xz). The control system can be written as x\ =b2{t)smxi

,

x2 =b2(t) cos xz ,

giving the integral curves of bi(t) X\(x) + 62(f) X2{x), Xi = -— , oxz

xz = bi(t),

(5-2)

where X\ and X2 are now

X2 = sin x3 -5 h cos x3 -z— • ox 1 dx2

(5.3)

294 The Lie bracket of both vector fields, d . d Xz = [Xi, X2] = •5—, smx3-r 0x3 ox\

d I-cos £3 -r— 0x2

d d • cos x3 — sin x3 -— ax\ 0x2 is linearly independent from Xi, X2, and {Xi, X2, X3} satisfy [X\, X2] = X3, [X2, X3] = 0 and [Xi, X3] = —X2, therefore closing on a Lie algebra isomorphic to the Lie algebra se(2) of the Euclidean group in the plane SE(2). This Lie algebra has a basis {ai, 02, 03} for which [ai, 02] = 03, [02, a 3 ] = 0 and [ai, 03] = —02. The adjoint representation takes the form ad(oi) =

/00 0 \ / 0 00\ 00-1 , ad(a 2 ) = I 0 0 0 , \0 1 0 / \-l 00/

ad(a 3 )

and as a consequence /l 0 0 \ exp(—vi ad(oi)) = I 0 coswi sinvi I , \ 0 — sin t>i cos vi J exp(—V2 ad(o2)) = 1

/ I 00\ 0 10),

exp(—v3 ad(os)) =

If we write the solution of the associated problem in the group SE(2) as the product g(t) — exp(—v\{€)a{) exp(—^2(^)02) exp(—v 3 (t)a 3 ), then (4.3) reduces in this case to vi ai+t>2 exp(—vi ad(ai))a2+t)3exp(—vi ad(ai))exp(—V2 ad(a2))o3 = 61 01+6202 • Performing the operations results the system vi = 61 ,

V2 = 62 c o s « i ,

i)3 = 62 sini>i,

(5.4)

with initial conditions i>i(0) = «2(0) = 113(0) = 0. Denoting Bi(t) = / 0 61 (s) ds, the solution is vi(t) = Bi(t),

v2(t) = I b2(s) cosBi(s)ds, Jo

v3(t) = I 6 2 (s) sinBi(s) ds . Jo (5.5)

An action of SE(2) on the original manifold R2 x S 1 such that the associated fundamental vector fields be Xi, X2, X3 is obtained by simply composing their integral flows. We parameterize the group SE(2) by the second class canonical coordinates obtained with the same factorization order as the one of the solution curve in the group, i.e. by three real numbers (9, o, b) with the composition law (9, a, b)(9', a', 6') = (9 + 9', a' +acos9' - bsinfl', b' +bcos9'+ asm9'). Therefore, the above mentioned action will read as $((#, o, b), (xi, X2, x3)) = (xi + bcosx3 + a s i n x 3 , X2 — bsinx3 + a cos2:3,2:3 + 9).

295 As a consequence, the general solution of (5.2) is $((«1, V2, V3),(X10, X20, X30)) = (xio + v3 cos X30 + V2 sin X30, X20 — v3 sin X30 + i>2 cos X30, X30 + « i ) , where vi = vi(t), V2 = V2(t) and v3 = v3(t) are given by (5.5). The direct integration of (5.2) gives exactly the same result. On the other hand, the vector fields X2 and X3 commute, so there exist coordinates (2/1,2/2,3/3) such that X2 = d/dy2 and X3 = d/dy3. Then, 1/2 will satisfy -X22/2 = 1 and -X32/2 = 0, and similarly 2/3 is such that X22/3 = 0 and X3y3 = 1. Particular solutions are 2/2 = Xi sinx3 + X2 COSX3 ,

2/3 = xi COSX3 — X2 sinx3 ,

which can be completed with y\ = x3. In this new coordinates X i takes the form Xi = -r 1-2/35 02/1 oyi

2/2-5— • oy3

Needless to say, these vector fields have to satisfy [Xi, X2] = X3l [X2, X3] = 0 and [Xi, X3] = —X2, as can be checked immediately. The control system of interest will be now the one giving the integral curves of the time-dependent vector field h(t)X!(y) + b2(t) X2(y), that is, yi=bi{t),

y2 = bi(t)y3 + b2(t),

y3 = - 6 i ( t ) y 2 •

(5.6)

The associated system in SE(2) is again (5.4). Now, X i , X2 and X3 can be regarded as the fundamental vector fields of the action of SE(2) on R 2 x 5 l given by /1 0 0 \ /yi\ / e ${{9, a, 6), (2/1, 2/2, 2/3)) = I 0 cos9 sinO I I y2 I + J a cos 6 + b sin 9 \ 0 — sin 9 cos 9 J \ 2/3 / \b cos # — a sin 6 Therefore, the general solution of (5.6) is given by $((wi, t>2, ^3), (2/10, 2/20, 1/30)), i.e. 2/i = 2/io + «i ,

1/2 = 2/20 cos t»i + 2/30 sin i>i + V2 cos ui + U3 sin vi , 2/3 = 2/30 cos «i — 1/20 sin v\ + v3 cos v\ — V2 sin «i , where v\ = i>i((), V2 = «2(<) and W3 = v3(t) are given by (5.5). However, the direct integration of (5.6) is, from the computational viewpoint, slightly more involved than the integration of (5.4).

296 5.3

Lie systems Hamiltonian

of elastic problem of Euler and related systems

integrable

This example is inspired by the recent works of Prof. Jurdjevic on the study of integrable Hamiltonian systems on homogeneous spaces of constant curvature embedded in a three dimensional Euclidean space, and the so called elastic problem of Euler n>12>13>14. These problems have important generalizations in 15 . However, we will treat here only the aspects of the problem related with Lie systems. The configuration space will be now R 3 , where we take coordinates (xi, x 2 , X3). The control system of interest is ii =-bi(t)x2-b2(t)x3,

±2 = bi(t)x\ + b3(t)x3 ,

£3 = e(b2(t)xi -

b3(t)x2), (5.7)

where e = ± 1 , 0 . This control system gives the integral curves of the time-dependent vector field b^t) Xi{x) + b2(t) X2(x) + b3(t) X3(x), where v

Xi = xi

d 0x2

d x2 - — , ox\

X2 = exi-

d dx3

d x3 -—, dxi

X3=x3-

d dx2

d ex2-—• ox3 (5.8)

These vector fields satisfy the Lie brackets [Xi, X2] = X3, [X2, X3] = eX\ and [X3, Xi] = X2, hence isomorphic to the Lie algebra ge of the Lie group G e , given by Go = S £ ( 2 ) , Gi = 5 0 ( 3 ) and G_i = SO(2,1). Therefore, the case e = 0 essentially reduces to the previous example. We take a basis {01, 02, 03} of ge in which the Lie products read [01, 02] = 03, [02, 03] = eai and [03, ai] = 02. Then, the adjoint representation is /00 0 \ ad(ai) = 0 0 - 1 ,

ad(a 2 ) =

\oi 0 I

/ 0 0 e\ 0 0 0 , ad(o 3 )

\-100y

/0-e0\ = 1 0 0 .

\o 0 0/

We define the signature-dependent trigonometric functions (see, e.g., 2 ) Ce(x), andT e (a;) by ( cos x Cc(x)=\l coshx

e= 1 e= 0 e = —1

{

( six sin x

Sc(x)=\x

^ ssii inhx

which have the properties Ce(x + y) = Ce(x)Ce(y)

e= 1 e= 0 e = —1

Se(x)

_ , . e(X)

T £ (x) =

— tSc(x)Sc(y),

e

^ '

Se(x + y)

297 Ce(x)Se(y)

+ Se(x)Ce(y)

and Ce2(x) + t S ( 2 ( i ) = 1. Then, we have

/I 0 0 \ exp(—t^i ad(oi)) = I 0 cosvi sintn I , \ 0 — sin vi cos vi J exp(—v2 ad(a2)) = I

0

1

0

J ,

\Se(v2) 0 Ce(v2) J exp(-«3ad(o3)) =

( Ct(v3) -Sc(v3)

V

e5«(«3)0\ C,(v3) 0 .

0

0

1/

Writing the solution of the problem associated to (5.7) in the group Ge as the product g(t) = exp(—vi(t)ai) exp(—v2(t)a2) exp(—v3(t)a3), and using (4.3), we obtain the differential equation system for vi(t), v2(t) and v3(t): vi = bi + eTe(v2)(b3 cosui + 62 sini;i), v2 = b2 cos v\ — 63 sin vi , 63 cos v\ + b2 sin v\ V3 = Cc(v2) with ui(0) = v2(0) = 1)3(0) = 0. If we take instead the factorization g(t) = exp(—v2a2) exp(—1)303) exp(—tnai), we obtain Vl =

6 i a ( f 2 ) + e635e(w2)

c&T)

'

v2 = b2 + Te(v3){b! Cc(v2) + i>3 = b3Cc(v2) -

eb3Se(v2)),

biSe(v2);

and the factorization g(t) = exp(—11303) exp(—uiai)exp(—v 2 a 2 ) gives vi = biCe(v3) -eb2Sc(v3)

,

v2 = (b2 Ce(v3) + 6i5 £ (?;3))secwi , v3 =b3 + (b2Ce(v3)

+biSe{v3))ta,nvi,

both of them with the initial conditions ui(0) = v2(0) = v3(0) = 0 as well. The other three remaining possibilities give rise to similar equation systems. For e = ± 1 the group Ge is not solvable and none of the previous systems can be integrated by quadratures in a general case. For the particular case treated by Prof. Jurdjevic we must put (with our notation) 61 (t) = 1, b2(t) = 0 and b3(t) = k(t). In his work a comprehensive study of certain related Hamiltonian systems is carried out, but this is out of the scope of this article. We refer to n > 1 2 ' 1 3 ' 1 4 ' 1 5 for a detailed exposition.

298 5.4

Brockett's

systems

There are certain control systems in the Control Theory literature known as Brockett's systems, see e.g. 4 ' 1 8 , which are known to be related with the tridimensional Heisenberg group H(3). We will study this fact with detail. The first form of Brockett's system we will consider is the control system in R 3 with coordinates (x, y, z) x = bi(t),

y = b2(t),

z = xb2{t) - ybi{t),

(5.9)

see, e.g., . This system gives the integral curves of the time-dependent vector field b1(t)Xi(x) + b2(t)X2(x), with Xi = £ - y £ and X 2 = ^ + z £ . The Lie bracket X3 = [Xi, X2] = 2 ^ is linearly independent from X\, X2, and {X\, X2, X3} close on the Lie algebra defined by [Xi,Xt]

= Xa,

[Xi,X3]

= 0,

[X2,X3]

= 0,

(5.10)

isomorphic to the Lie algebra h(3) of the Heisenberg group H(3). The Lie algebra h(3) has a basis {ai, a2, 03} for which the Lie products are [01, a2] = 03, [ai, 03] = 0 and [a2, 03] = 0. The adjoint representation of f)(3) is then /0 0 0 \ ad(ai) = 0 0 0 ) ,

ad(o 2 ) = |

0 0 0 ) , ad(a 3 ) =

and therefore exp(—vi ad(ai)) =

/ I 0 0\ 0 1 0 , \0-wi 1/

exp(—v2 ad(o 2 ))

exp(—V3 ad(a3)) = Id . Writing the solution of the associated problem in H(3) as the product of exponentials g(t) = exp(—ui(£)ai)exp(—v2(t)a2)exp(—1)3(^)03) and applying (4.3), we find the differential equation system vi = bi ,

v2=b2,

i>3 = b2vi,

(5-11)

with initial conditions «i(0) = ^2(0) = ^3(0) = 0. The solution can be immediately found: vi(t)=

[ bi(s)ds, Jo

v2(t)=

/ b2(s)ds, Jo

v3(t) = / b2{s) / Jo Jo

bi(r)drds. (5.12)

The elements of the Heisenberg group H(3) can be represented by three real numbers (a, b, c) with the composition law (a, b, c)(a ,b,c) = (a + a,b + b,c +

299 c' + 6a'). In a similar way as in previous examples, the vector fields Xi, X2 and X3 can be shown to be the fundamental vector fields of the action of H(3) on R 3 given by $((a, b, c), (x, y, z))

Then, the general solution of (5.9) is * ( ( « i , v2, V3), (xo, yo, zo)), i.e. x = xo + «i, y = yo + V2 and z = zo + xo«2 — J/o«i + 2t>3 — «i«2, where «i = v\(t), V2 = V2{t), and v3 = v3(t) are given by (5.12). The direct integration of (5.9) gives the same result. Other form of Brockett's system 18 is the control system in R 3 with coordinates (*, V, z) i = tn(t),

y = b2(t),

z = -bi(t)y.

(5.13)

This system gives the integral curves of the time-dependent vector field 61 (t)X\ + b2{t)X2, where X x = £ - y§-z and X2 = ±. If we take X3 = [Xly X2] = £, it is easy to see that {Xi, X2, X$} satisfy the Lie brackets (5.10). Moreover, they can be regarded as the fundamental vector fields of the action of H(3) on R 3 given by

*((a, b, c),(x, y, z)) =

The general solution of (5.13) is now $ ( ( « i , ^2, U3), (%o, yo, zo)), i.e. x — xo + vi, y = yo+V2 and z = zo — yovi+V3 — viV2, where vi = vi(t), V2 = V2(t), and V3 = V3(t) are given by (5.12). Direct integration of (5.13) gives the same result. Finally, another realization of the same system can be obtained from the natural action on R 3 of the representation of the group H(3) as upper triangular 3 x 3 real matrices. Taking coordinates (x, y, z) on R 3 , this action reads as

H(a, b, c), (x, y, z)) =

The associated fundamental vector fields are now X\ = z-g-, X2 = y£ and X3 = z£, which satisfy (5.10) as well. Then, the general solution of the control system giving the integral curves of 6i(t)Xi + &2(£)X2, that is, x = b2(t)y,

jr = 6 1 (t)z,

i = 0,

(5.14)

is $ ( ( m , V2, V3), (xo, yo, z0)), i.e. x = xo + yovi + z0V3, y = yo + z0vi and z = z0, where v\ = vi(t), V2 = V2{t) and 113 = V3(t) are given by (5.12). Once again, the direct integration of (5.14) gives exactly the same result.

300 5.5

Front-wheel

driven kinematic

car

This example is based on a model for an automobile proposed by Murray and Sastry . The configuration space is R2 xS1 xS1, where we take coordinates (x, y, , 0). The control system is (we take their constant Z to be 1) x = ci(t),

y = a(t)t&n0

,

<$> — 02(f),

8 = ci(t) tan cf> sec 9,

(5.15)

which gives the integral curves of the time-dependent vector field c\(t)X\ + c2(i)X2, where X\ = - 5 - + t a n 0 — +tan s e c 0 — , ox oy 09

X2 = — . dtp

Taking Lie brackets we can try to find the minimal Lie algebra containing these two vector fields. It is not difficult to see that such an algebra is infinite-dimensional so the system is not a Lie system. However, it can be approximated by a Lie system as follows. Making the local feedback transformation proposed in 21 (although it seems that there are some misprints there) xi=x,

X2 = sec 3 9 tan <j>, 2:3 = tan 0,

c\=b\

C2 = — 3sin <j>sec 9sin9bi+

x\ = y, cos 8 cos <j>b2,

we obtain the control system on R 4 , with coordinates (xi, X2, X3, Xi) i\ = 61 ,

£2 = 62,

xs = X2b\,

±4 = 2:361,

(5.16)

which is a system in chained form, according to the terminology of 2 1 . It is interesting to note that in this case, the system (5.16) can be essentially obtained approximating all of the trigonometric functions appearing in (5.15) by their zero order Taylor series, see also 18 . Then, the control system (5.16) gives the integral curves of the vector field bi(t)Xi + b2(t)X2, where now Xi = ^ + x 2 ^ + x 3 g f j and X2 = g f j . Taking the Lie brackets d X3 = [Xi, X2] = - -r— , 0x3

d Xi = [Xi, X 3 ] = - — , 0x4

we obtain two other linearly independent vector fields such that {Xi, X2, X3, X4} close on the Lie algebra defined by [Xu X2\ = X3 ,

[Xi, X3] = X4 ,

(5.17)

being all other Lie brackets zero. The corresponding abstract Lie algebra has a basis {ai, a2, 03, 04} for which the only non-vanishing Lie products read as [01, 02] = 03,

301 [oi, 03] = 04. The adjoint representation takes the form

0000\ 0000 0100 00 1 0 /

ad(ai)

0 000\ I 0 000

AI \ ad fl2)=

(

I -1000 0 0 00/

/ 0 000\ 0 000 ad(a3) = 0 000

/oooo 0000

ad(a4) =

V-10 1 0/

0000

\0000,

and therefore

vl

exp(—v\ ad(ai)) = Id —vi ad(ai) + — ad(ai) o a d ( a i ) , exp(—w 2 ad(a2)) = Id— V2 ad(o 2 ), exp(—V3 ad(o3)) = Id— 1)3 ad(a3),

exp(—V4 ad(a4)) = Id .

Writing the solution of the problem associated to (5.16) in the Lie group as g(t) = exp(—viOi)exp(—i;2ffl2)exp(—^303) exp(—1)404), and applying (4.3), we obtain the system

Vl = 6 l ,

i)2 = 62 ,

V3 = b2V\ ,

1)4 = 62

vl

(5.18)

with initial conditions tn(0) = 1)2(0) = 1)3(0) = 114(0) = 0, which is easily integrable by quadratures. Denoting B\{t) = fQbi(s)ds, £?2(0 = fQb2(s)ds, the solution reads vi(t) = B1(t),

V2{t) = B2(t),

v3(t)=

f

b2(s)B1(s)ds,

Jo

Vi{t)

-\f.«

s)BJ{s)ds.

(5.19)

The Lie group whose Lie algebra is the one described above is four dimensional, and its elements can be represented by four real numbers (a, b, c, d) with the composition law (a, 6, c, d)(a, b', c , d!) = (a + a , b + b', c + c + ba, d + d' +ca' + b(a'f / 2 ) . Now, the vector fields Xi, X2, X3 and X4 can be shown to be the fundamental

302 vector fields of t h e action t h e previous Lie g r o u p on R 4 given by

$ ( ( a , b, c, d ) , ( x i , x 2 , x 3 , x 4 ) ) =

n0L 01 00 00 \

fxA X2

0 a 10

X3

\0^-al) /

\x4J

a b ab —c

-

\

: + dJ

V

T h e n , t h e general solution of (5.16) is $ ( ( w i , v2, v3, i>4), (xio, X20, £30, X40)), i.e. Xl = Xio + Ul ,

X2 = X20 + «2 ,

^3 = X30 + X20«l + «ll>2 — V3 , V? v\v2 X4 = X40 + :E30«1 + £20"^" H ^~

' V\V3 + Vi ,

where vi = vi(t), V2 = V2(t), V3 = V3(t) a n d V4 = V4(t) are given by (5.19). Direct integration of (5.16) gives t h e s a m e result. O t h e r realization of t h e control s y s t e m (5.16) can b e o b t a i n e d w h e n one considers t h e n a t u r a l action o n R 4 of a r e p r e s e n t a t i o n of t h e previous Lie g r o u p as u p p e r t r i a n g u l a r 4 x 4 real m a t r i c e s . Taking again c o o r d i n a t e s ( x i , X2, X3, X4) on R 4 , t h i s action r e a d s

$ ( ( a , b, c, d), ( x i , x 2 , x 3 , x 4 ) )

/16 c Ola 00 1 \000

d \ Xl ( \ ^ X2 0 X3 1 ) \X4j

T h e associated f u n d a m e n t a l vector fields are now Xi = X3Tp— + X4-J*—, X2 = X2 •dxi

>

X3 = x3^r a n d X4 = X4-^-, whose non-vanishing Lie b r a c k e t s are given by (5.17). T h e control s y s t e m giving t h e integral curves of t h e vector field b\(t)X\ + 62(4)^2 is X1=62(*)X2,

X2=bl(£)x3,

X3 = 6 l ( * ) x 4 ,

X4 = 0 ,

(5.20)

whose general solution is j u s t 3>((wi, V2, V3, V4), (xio, X20, X30, X40)), i.e. Xl = Xio + l20«2 + X30H3 + £40^4 1

X2 = X20 + X30UI + £40 —

X3 = 2:30 + X40W1 ,

X4 = X40 ,

where vi = vi(t), V2 = V2(t), V3 = V3(t) a n d V4 = V4(t) a r e given by (5.19). s a m e result c a n b e o b t a i n e d i n t e g r a t i n g directly (5.20).

The

303

Acknowledgments A.R. is supported by FPI Grant PB96-0717 of the Spanish Ministerio de Educacion y Cultura. A.R. also thanks the organizers of the Velfest, GCTAA-2000 for their kind invitation and support. Support of the Spanish DGES (PB96-0717) is also acknowledged.

References 1. J. Baillieul and J.C. Willems Eds., Mathematical Control Theory, (Springer Verlag, New York, 1999). 2. A. Ballesteros, F.J. Herranz, M.A. del Olmo and M. Santander, Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries, J. Phys. A: Math. Gen., 26, 5801-5823 (1993). 3. R.W. Brockett, System theory on group manifolds and coset spaces, SIAM J. Contr., 10, 265-284 (1972). 4. R.W. Brockett and L. Dai, Nonholonomic kinematics and the role of elliptic functions in constructive controllability, in Nonholonomic Motion Planning, Li Z.X. and Canny J.F. Eds., (Kluwer, Norwell, MA, 1993). 5. J.F. Carinena, Sections along maps in Geometry and Physics, Rend. Sem. Mat. Univ. Pol. Torino, 54, 245-256 (1996). 6. J.F. Carinena, J. Grabowski and G. Marmo, Lie-Scheffers systems: a geometric approach, (Bibliopolis, Napoli, 2000). 7. J.F. Carinena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta App. Math., in press. 8. J.F. Carinena and A. Ramos, Integrability of the Riccati equation from a group theoretical viewpoint, Int. J. Mod. Phys., A 14, 1935-1951 (1999). 9. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, (Academic Press, New York, 1978). 10. V. Jurdjevic, The geometry of the plate-ball problem, Arch. Rat. Mech. Anal, 124, 305-328 (1993). 11. V. Jurdjevic, Optimal control problems on Lie groups: crossroads between geometry and mechanics, in Geometry of Feedback and Optimal Control, Jakubczyk B. and Respondek W. Eds., (Marcel Dekker, New York, 1993). 12. V. Jurdjevic, Non-euclidean elastica, Amer. J. Math., 117, 93-124 (1995). 13. V. Jurdjevic, Geometric Control Theory, (Cambridge University Press, New York, 1997). 14. V. Jurdjevic, Optimal control, geometry, and mechanics, reference 1 , 227-267. 15. V. Jurdjevic, Integrable Hamiltonian systems on Lie groups: Kowalewski type, Ann. Math., 150, 605-644 (1999). 16. V. Jurdjevic and H.S. Sussmann, Control systems on Lie groups, J. Diff. Eq., 12, 313-329 (1972). 17. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I,

304 II. (Wiley, New York, 1963). 18. N.E. Leonard, Averaging and Motion Control of systems on Lie Groups, Ph. D. Thesis, (University of Maryland, 1994). 19. S. Lie, Vorlesungen uber continuierliche Gruppen mit geometrischen und anderen Anwendungen, revised and edited by Dr. G. Scheffers (B.G. Teubner, Leipzig, 1893). 20. V. Manikonda, P.S. Krishnaprasad and J. Hendler, Languages, behaviors, hybrid architectures, and motion control, reference 1, 199-226. 21. R.M. Murray and S.S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Auto. Control, 38, 700-716 (1993). 22. H. Nijmeijer and A.J. van der Schaft, Nonlinear dynamical control systems, (Springer Verlag, New York, 1990). 23. J. Wei and E. Norman, Lie algebraic solution of Linear Differential Equations, J. Math. Phys., 4, 575-581 (1963). 24. J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc, 15, 327-334 (1964). 25. P. Winternitz, Lie groups and solutions of nonlinear differential equations, in Nonlinear Phenomena, K.B. Wolf Ed., Lecture Notes in Physics, Vol. 189, (Springer Verlag, New York, 1983).

305

FROM THE GEOMETRY TO T H E ALGEBRA OF NONLINEAR OBSERVABILITY

SETTE DIOP Laboratoire des Signaux & Systemes CNRS-Sup elec-Univ. Paris Sud Plateau de Moulon 91192 gif sur Yvette cedex France E-mail: [email protected] This contribution goes along the lines of the belief that nonlinear observability has a deep geometric content. We try to set up that geometry, its algebraic translation all the way down to the algorithms needed to do some of the computations. The basic geometric picture of observability is that of a projection map, that of the system trajectories onto the data trajectories. Observability is then seen as the condition for this map to be finite in the sense that its fibres are generically finite. It is this definition which easily passes to differential algebraic geometric characterizations. Next, most of the picture is kept by defining the notion of singular observation data, that is, the special data for which the generic observability property is lost. The work is incomplete at least in that we hardly touch the so-called real case where we have to leave the abstract differential algebraic geometry to enter the differential real algebraic geometry. This is under investigation.

Dedicated to Velimir Jurdjevic 1

Introduction

A system x = f(t,u,x), y = h(t,u,x), is a Um the Um

{

>

set X of trajectories (u,x,y) satisfying equations (1) in some affine space x Un x Up, where U is the function space where the control, the state, and output take their values. An external trajectory 1 6 (u,y) is an element of x Uv for which their is at least an x such that (u, x, y) £ X. The set of external trajectories is called the external behavior 1 6 of the system and denoted by Xu
306 there is at least one such a state x. The word determine rather refers to the number of possible values of x. Formally, we have a projection map IT of the set X onto the set Xu,y which sends a trajectory (u, x,y) into its corresponding external trajectory (u, y) according to the following picture.

Figure 1. Variety projection picturing observability notion.

Given an external trajectory (u,y) in some time interval we shall be in one of the following three situations. The first situation is when -K~X(U, y) contains infinitely many elements. The second one is when 7r _1 (S,y) is finite with cardinality greater than 1. The last one is when 7r - 1 (u,|/) is a singleton. The word determine in the above setting of the observation problem also has some practical content which refers to the ability to explicitly compute or approximate all the possible values of x corresponding to (u,y). This want leads to the statement that the observation problem is solvable, we say that the system is observable, if TT"1 (u,y) is finite. Finiteness, or even cardinality 1, of 7r _1 (u, 1/) for every external trajectory (u,y) is actually the ultimate need from a practical standpoint. But simple characterizations of observable systems requires a weaker condition. We shall thus rather qualify a system as observable if the projection map IT is generically finite. Denoting by (u,y) a generic external trajectory this means that we actually require 7r -1 (u,2/) to be finite for the system to be observable. The eventual external trajectories (u, y) for which ir~l{u, y) is infinite are then considered as singular ones for the observability of the system.

307

To have a theory of observability we need to specify the class of function spaces U over which the trajectories take place. The differential algebraic geometric approach of observability that is to be presented here aims at providing a mathematical sound language to the previous intuitive interpretation of observability. Observability theory for nonlinear systems has been a long term research interest for many authors. Its pertinence being sufficiently explained by the germaneness of estimation problems in control system designs. Following the work of R. E. Kalman for linear systems, the notion of distinguishability of state space points has been used to define observability. Referring to our previous abstract presentation of the observation problem we may say that, the classical theory would rather state observability as the condition where 7r _1 (u, y) contains only 1 element for every external trajectory (u,y). The reader may refer to 7 for more details on this approach. The book 6 by J. P. Gauthier and I. A. K. Kupka provides a thorough study of nonlinear observability and observer design. The differential algebraic approach of nonlinear observability dates back to 1986 with a paper by J. F. Pommaret n . But the approach was first explicitly developed in 5 ' 4 . The present work is an extension of the previous two papers. More specifically, we consider general systems described by Pi(w,z,0=0, Q(w,z,O^0,

* = 1,2,...,

,s {Z)

where the Pi's and Q are finitely many polynomials in w, z, £ and their derivatives. The variable w stand for the data (in the classical observation problem, w is (u, y), the input and the output), z is the variable to be observed (or estimated), and £ is an extra variable which may eventually be present in the system's description and which is neither part of data nor being observed. The coefficients of the system's equations are supposed to be in a differential, field, k, of functions of one variable, the time, whose derivation is the standard derivation with respect to the time. The above mentioned function space U over which the system variables take their values is here a differential algebraic closure, k, of k. The set, X, of trajectories of system (2) is thus a subset of an affine space over k. A basic assumption we make throughout this approach is that X is irreducible. Its projection XWjZ along the variables w and z (i.e. the set of (w, 1) in the ambient affine space for which there is £ such that (w, ~z, Q £ X) is thus irreducible, too. Similarly for the projection, Xw, of Xw z along the variable z. Let •n :

308

%w,z -> Xw the latter projection map. Let XWjZ and Xw be the respective closures of XWiZ and ^ for the differential Zariski topology on the ambient affine space. These sets have generic points, that is, points which are dense in the respective set. Let (w, z) be a generic point of XWtZ. Our formal definition of observability reads as: z is observable with respect to w if n~1(w) is finite. The previous geometric definition of observability now has a purely algebraic translation: z is observable with respect to w iff k(A'tU]2} is algebraic over k(Xw) where the latter are differential field extensions of the field of coefficients. Computationally, z is observable with respect to w iff for each component, Zi of z there is a polynomial equation Hi(Zi,w,w,...)=0

(3)

in Zi, and finitely many time derivatives of the data w, with coefficients in k. It turns out that the situation where the Pi's and Q are not necessarily polynomial may be coped with by merely relaxing the polynomial requirement on Hi (this function still has to depend on only finitely many data time derivatives). But we do not enter this generalization. This definition raises at least three basic questions. First, what if the data assume some value w such that all existing relations (3) for Z{ degenerate into the trivial equation 0 = 0? Second, what if all such relations as (3) are known to have more than one solution in zp. Third, does the definition require more regularity of the data than the latter are in effect? The first question is related to the singularity of the observability notion with respect to the data. We shall address it precisely in 2.3. The second question is about a so-called local character of the observability of the system we have at hand. A partial answer is provided through the notion of rational observability addressed in 2.2. And the third question asks for some extended version of the present differential algebraic theory. We discuss it in 2.4. We have tried to make this contribution as self contained as possible. We thus have provided appendix sections giving key details on the differential algebraic tools necessary to read the paper for someone not familiar with them. Most of the technical differential geometric notions appearing in the next two sections are explained in the appendices. 2

The differential algebraic geometric approach

In this approach we call (differential) (algebraic) system with s variables, and with coefficients in k a proper differential quasi-affine variety X C k defined

309

over k where k is a differential closure of k. The differential dimension of X over k is the (minimum) number of input components. An input is a differential transcendence basis of k(X) over k. Once an input is chosen the other components of the system variable are differentially algebraic over k{u). In observation problems, the system variable is partioned into the data, or observations, w = wi,... ,w^, the variable being observed (or estimated) z — z\,... ,zn and the remaining variables, £. In what follows we assume n > 1. In the classical observation problem, the data consist of the control u and the measurements y. The present approach allows to consider more general observation problems. When the variable £ is present, we consider the projection Xw
Irreducibility of rational state systems

It is important to be able to deal with systems (1) where / and h are rational functions of u and x with coefficients in some differential field k. Such systems read as

Xi

Pi{u,x) qi(u,x) fj(u,x) 9j(u,x)

(l
where u stands for ux, u 2 , . . . ,um, and Pi, qt, fi and p; are differential polynomials of order zero in x with coefficients in k. Written by means of equations

310

and inequations the previous system is equivalently described by the following ' qi(u,x)xi=pi(u,x) 9j («, x) Vj = fj (u, x) <

n

(l
p

Y[qi(u,x)Y[gj(u,x) , i

^0.

i

The set X of trajectories of this system is defined to be the elements (u, x, y) £ k x k x k which satisfy Pi{u,x,y) IA(u,x,y)

= 0 ^ 0,

{l
where J^ is the multiplicative set generated by q\ (1 < i < n) and gj (1 < j < p), and where k is a differential closure of the differential field of coefficients k, and where Pi(U,X,Y)

= qi(U,X)X{i1)

Pn+j (U, X, Y) = fj(U,X)-

-Pi(U,X) 9j(U,

(1 < i < n),

X)Yj

(l<j
We need to establish the irreducibility of X so that (1) defines a system in the sense of the present approach. Having done this we shall call such systems rational state systems. Let k {[/, X, Y} be equipped with the ranking {{UuU2,...,Um},{X1,X2,...

,X„,y1,Y2,...

,YP}},

see Notation 1. Then the set A consisting of Pi,P2, • • • , Pn+P is readily an orthonomic autoreduced set. By Lemma 2.10, p = [A]: I™ is a prime differential ideal with characteristic set A. Let p define the affine variety X', (u,x,y) € X, and P € p . There is v e !N n + p such that I\P € [A] so that I^(u,x,y)P(u,x,y) = 0. Since (u,x,y) £ X implies that I^(u,x,y) ^ 0, we have P(u,x,y) = 0. Therefore X C X'. Next, let X" consist of the elements of X' which do not annihilate IVA. It is clear that X C X" because, as we just have proved, X is contained in X'. Conversely, let (u, x,y) £ X". Since Pj 6 p we have Pi(u,x,y) = 0. Therefore, X" C X. It results that X = X", that is, X is an open subset of X'. The fact that A is a characteristic set of p also implies that

31 1

lAf][A] =
Definitions and properties

Definition 2.3. Let X be a system with variables w, z, and (, and with coefficients in k. The variable z is said to be (algebraically) observable with respect to w if the projection map TT : Xw>z —> Xw (sending every trajectory (w,~z) of XWyZ into the corresponding observation w) is generically finite. If z is observable with respect to w then the degree of 7r is called the observability degree of z with respect to w, and is denoted by d^z. The variable z is said to be rationally observable with respect to w if it is observable with respect to w with observability degree one. For state systems of the form (1) we say that the system is observable if its state variable is observable with respect to u and y. Theorem 2.4. Let X be a system with variables w, z, and C,, and with coefficients in k. The variable z is observable with respect to w if, and only if, k(A'u,]Z) is algebraic over\s.{w). This follows directly from Lemma 1.4. Theorem 2.5. Let X be a system with variables w, z, £, and £, and with coefficients in k. If Q is observable with respect to (w,z), and z is observable with respect to w then £ is observable with respect to w. This transitivity of the observability property follows immediately from that of the algebraicity property. Theorem 2.6. Let X be a system with variables w, z, and £, and with coefficients in k. If z is observable with respect to w we have d°wz = A°wzl • d°WZlz2

•w,z\,Z2,... ,zn-izn J

312

where d^, Zl
where hi and qi are differential polynomials with coefficients in k. The proposition is thus immediately proved. For example, the following system

x2=u, y = xi,

(4)

is rationally observable since xi=y

and

x2 = - — 2u

But the system Xi

— —X2 ,

x2 = ux2 , y =

(5)

xi,

is not rationally observable since xi = y

and

x\ =

-y,

313

and there is no means to reduce the degree of the observability condition of x2. For rational state systems observability is equivalent to the fact that the number of state components is equal to the order of the system. To prove this we need the following result. Lemma 2.8. For the system (1) the (nondifferential) field extension k(u, x, y) of k(u) is purely transcendental. The (nondifferential) field k(X) =

k(u,x,y)

= k(u)(a; 1 ,a;2,... ,xn,ii,...

,xn,...

, j / i , . . . ,yP,yi,---

,yP,---)

is equal to k(u)(xi,X2,• • • ,xn). The result then follows from Lemma 2.10. Proposition 2.9. For a rational state system X the following two conditions (i) X is observable, (ii) the number of components of the state variable x in (1) is equal to the order, oX, of X, are equivalent. The order of X is defined to be the transcendence degree of k(^f„i2/) = k(u,y) over k(u). We have dk(«>k
+

d^{u)k(u,y).

By Lemma 2.8, d k , ,k(u)(x) is equal to the number of state components, while by the observability assumption, d k , >k(u,x,y) = 0. This proves the proposition. 2.3

Regular observability

Let X be a system with variables w, z, and £, and with coefficients in k. Recall the observation projection map ir : XWtZ —> Xw sending any trajectory (uJ,z) onto W. We assume z to be observable with respect to w, that is, the inverse image of the generic point w of Xw is with finite cardinality. It is a fact that, for special observations w, 7r~1(w) may contain infinitely many elements, leading to a singularity of our generic notion of observability. An example of such situations is the following

314

±i = xxx2 , x2 = u + x2 , y = xi.

(6)

x is observable with respect to u, y since x\—y

and x2 = — • V But in practice, in any time interval where y is identically zero (or, merely, small), the observability of x2 is singular in the sense that it is lost. Definition 2.10. Let X be a system with variables w, z, and £, and with coefficients in k, and such that z is observable with respect to w. An observation w £ Xw is said to be singular for the observation of z with respect to w ifTr~1(w) is infinite. Observations w € Xw which are not singular are called regular. The variable z is said to be regularly observable with respect to w if there is no singular observation for its observability with respect to w. For instance, system (4) is regularly observable. R e m a r k 2.11. Our definition of singular observations certainly copes with basic needs in practice. However, since the finiteness requirement on the fibres of the projection map n is no more generic we have to care about the conditions on passing from the geometric objects to corresponding algebraic ones. Recall that the differential algebra k {w} refers to the closure of Xw which may strictly contain the latter set. Likewise, the differential algebra k {w, z} refers to the closure of XWiZ which, also, may strictly contain the latter set. Given w € Xw it may be that 7r_1 (w) is finite when it is considered as a map XWtZ —> Xw while n~1(w) is infinite for ir : XWjZ —»• Xw. But, of course, for W £ Xw, if n~1(w) is finite for -K : XWjZ —> Xw then 7T-1(wJ) is finite, too, for IT : Xw Xw. The algebraic characterization of singular observations we are seeking here is in terms of properties o / k { w } and ]t{w,z}. Such conditions will be sufficient only. We have the following partial result whose proof is immediate. T h e o r e m 2.12. 3 Let X be a system with variables w, z, and £, and with coefficients in k. The variable z is regularly observable with respect to w if z is integral overk{w}. Integrality means that each component of z is a solution of a monic polynomial equation (one with leading coefficient 1) with coefficients which are differential polynomials (and not rational fractions) in u and y. The condition in the last theorem is not necessary as shown by the following example

315

±i —XiX^ +X2 , X2 = U + X2 , y = zi.

(7)

This system is observable given the equations for y and xi. We may see that its observability degree is actually 1. The variable x2 is not integral over \i{u,y}, but the system may be seen as regularly observable since when y is not identically zero we have y x\ + x2 - y = 0 while, when y is identically zero, we have x2=y

= 0.

The situation may be more complex as shown by the following simple example Xi = UX2 , (8)

x2 = xi,

y =differential xi. The system is not closed for the Zariski topology, its closure is described by

The system is observable since

x\—y

and

y

x2 = —, u x is not integral over k {u, y}, but the system is regularly observable since its external behavior is described by uy — iiy — u2y = 0 ,

316

so that there is no singular external trajectory for the observation of x. We proceed to extend the result in Theorem 2.12. Let k{w,z} be a differential algebra generated by the elements in w and z and let k {w, z} be an integral domain. An element £ of k {w, z} is said to be primitive over k {w} if it is a zero of a polynomial ad id + a d _i £ d _ 1 + • • • + a 0 = 0 such that (i) the aj's are in k {w}, (ii) the perfect differential ideal {ad(w),ad-i(w),... unit ideal.

,a0(w)} of k.{w} is the

We have the following result whose proof is also immediate but which seems to be the best one can get for the characterization of regular observability from the algebraic properties of the differential algebras associated with the system. Theorem 2.13. Let X be a system with variables w, z, and C, and with coefficients in k. The variable z is regularly observable with respect to w if z is primitive over\a{w}. This theorem allows to see that system (7) is regularly observable since X2 is primitive over k{u,y}. But the theorem also fails to see that system (8) is regularly observable. 2.4

Observability and data differentiability

The discussion we enter here is about the extension of this differential algebraic geometric theory of observability to systems whose variables are allowed to live in spaces of functions which, say, are not smooth as required by differential algebraic affine spaces. The discussion is mostly motivated by systems whose input u may be not smooth. In such cases we cannot blindly apply the usual rules on operations of differential algebras. To be more specific, let us consider the following simple linear example ±1 = X2 + u , ±2 = -xi - x2 ,

y = x\ + u. This system is observable since

(9)

317

x\ = y — u

and

x2 = y — ii — u.

If the control, u, is Lebesgue-measurable, but not continuous, say, u is piecewise continuous with finite jumps then, in the usual sense of derivatives, u is not differentiable, and the expression x2 = y - u — u is not valid. If u is not differentiable then we note that the observability of x 2 reads as X2 = Xi — U ,

given the first equation in (9). Since xi = y - u is differentiable, we rather write x2 in the following form x2 = (y-u)'

-u,

which evacuates the need to differentiate u, and, at the same time, sheds light on how the data differentiability question did enter the scene. It basically results from the application of the following rule (a + b)' =a + b. This rule is of course not applicable on function spaces which are not closed under derivation. In nonlinear examples we may encounter the other usual "distributivity" rule which may not apply in the derivation of the observability conditions: (a • b)' = ba + ab. For constant-coefficients linear system we may write down the observability condition without violating the differentiability of the observations. Theorem 2.14. For constant-coefficients linear systems (± = Fx + Gu, \y = Hx + Eu,

K

'

where F, G, H, and E are real matrices of appropriate sizes the observability of x with respect to u, y when u is taken over a differential closure o/k = H is equivalent to the observability of x with respect to u, y when u is taken over the

318

space of piecewise continuous real-valued functions. Moreover, the differential algebraic observability condition may be rewritten so as it does not explicitly invoke the proper derivative of u. The equations of y in system (10) define a finite type algebra generated by u, y and x that we denote by k[u,x,y]Q. We abuse notation by using the same symbols u, y and x since, here, the latter are rather the residues of corresponding indeterminates modulo the equations of the output (and not the equations of the whole system), but there will be no fear of confusion. Let X 0 be a maximal subset (in the sense of inclusion) of xi, X2, • •., xn which is algebraic over k[u,x,y]0. Let no be the cardinality of Xo- no > 1 since otherwise the system is not observable. If no = n then the theorem is proved. We next assume that 1 < no < n. To simplify the notations we assume that the members of Xo are x\, X2, •. •, xno. For each i < no we have zo,i — Xi = aiy + biU where ctj and fcj are real constant row vectors. By definition of the system the XiS are differentiable. We differentiate the Xj's in X 0 to obtain relations z\ = i 0 = Hix + E\u where ZQ is the vector with components .zo,i, 1 < i < no, Hi is a submatrix of F and E\ is a submatrix of B. The equations ( y = H x + Eu, \ z\ — H\x + Eiu, again define a finite type algebra generated by u, y, x and z\ that we denote by k[u,x,y,zi]0. Let Xi be the maximal subset ofxi, X2,..., xn which contains X 0 and which is algebraic over \a[u,x,y,zi]0. Let n\ be the cardinality of X i . We have n 0 < n\. If n\ = no then the system is not observable. If m = n then the theorem is proved. Next we assume 1 < n 0 < n\ < n. We repeat the previous construction to obtain an increasing sequence of integers 1 < no < ni < • • • < n which has to stop. This proves the theorem. This result clearly generalizes to nonconstant-coefficients linear systems as long as the coefficients are in a ring of functions where they have inverses. For nonlinear systems one may see that the same procedure outlined in the previous proof applies as long as, at each step, the state components which are algebraic over the algebra generated by u, y, x, z\, etc. are actually rational over the same algebra.

319 3

Computing

We have collected here some algorithms which may be used as computing devices in nonlinear observability. The first one is an observability test valid for polynomial systems, i.e., those defined by polynomial differential equations. fi{w,w,...

,z,z,...)

= 0,

i = 1,2,... .

This test reduces to a general rank condition which is a necessary and sufficient condition for the observability of z with respect to w. When specialized to the observability (of x with respect to (u,y)) of rational state systems we obtain a rank condition which is similar to (but, a priori, not the same as) the well-known observability rank condition in 7 . 3.1

A rank condition for polynomial systems

For polynomial systems, a quite general observability test may be derived in terms of the rank of a Jacobian matrix. The basic tools are (Kdhler) differentials for which we include the following brief account. Let R be a commutative ring with unit element. Let A be an R-algebra with structural morphism p : R —> A. Let M be an A-module. An Rderivation of A into M is a map D : A —> M which satisfies the following three axioms: (i) D(x + y) = Dx + Dy (x, y € A), (ii) D(xy) = yDx + xDy (x, y £ A), and (iii) D(p(a)) =

0(aeH).

The basic property of differentials of an R-algebra A is the following. There exist an A-module, usually denoted by O R (A) (or by Q A / R ) > and an R-derivation, C?A/R •' A —• J)R (A) (called the canonical R-derivation of A), such that, for any A-module M , any R-derivation D of A into M , there is one, and only one, A-module morphism D of H R (A) into M such that D = DdA/R (when no confusion may occur, the sub-index with d will be dropped). For example, let (T*) i6l be a family of indeterminates over the ring R. Let o be an ideal of R {(Ti)i€i\ and A = R [(Ti)iei] / o = R [(*i) ie i]. Then fiR(A) is the quotient of the free A-module with basis ((dtj) i e I ) by the submodule generated by the A-linear forms

320

Y,gjr(t)dti

(Pea).

The main fact we shall make use of is that, if R = k is a field (of characteristic zero), and if A = K is a field extension of k, then the transcendence degree d k K of K over k is equal to the dimension [ftk(K): K] of the K-vector space fik(K). More precisely, a family (ti)i€l of elements of K is algebraically independent over k if, and only if, the family (dti)iel of elements of fik(K) is K-linearly independent; and, for K to be algebraic over k ((ii)iei), it is necessary and sufficient that the K-vector space fik(K) is generated by (dti)i€l. In particular, let the field extension K of k be of finite type, generated by n,... ,r„ (that is, K = k ( r i , . . . ,rli)). Let Pi, P 2 , . . . , Pa be a set of generators of the ideal of definition of K over k, i.e., k

[TI.TS,

...

,T„]

= k [TltT2,...

,r„]/(Pi,P2,

...,P„)

(where the T"s stand for indeterminates). Then d k K = [fi k (K) :K] = v - r k K J ( P i , P 2 , . • • ,P„) where J ( P i , P 2 , . . . ,Pa) denotes the Jacobian matrix of P i , P 2 , . . . ,P f f , i.e., the matrix

\dPi (

0

where 1 < i < a, and 1 < j < v, and where rkK (Pi, P2, • • • , PQi -i2f'' _1 + • • • + aifi = 0, with minimal degree, a,, and with coefficients in k(w). Then i, verifies (atz^-1

+ (at - l)ai, Q i _izf i _ 2 + • • • + a u ) ii + a i ; Q i _ i 2 t a i _ 1 + • • • + a i , 0 = 0,

321

that is, ii is in k(u/)(zj) since, in characteristic zero, by the minimality of at , we have ctiz"'-1 + (a, - l ) a i i a j _ i z ? i - 2 + • • • + Oi,i ^ 0. Since k(w, z) = k(w) I \z\3') \\

J, our claim is proved by an immedi/ l<»
ate induction on j . Actually, we obtained a more general fact: for any given rj € 1N(1 < j < n), the algebraicity of z over k(w) is equivalent to that of

k(w)((z[h\4h),...,z^A \ \

)

'

/0<ji
over k(w). Notation 1. If z = (zj)i<,-< n and r = (rj)i<j< n € IN" then the family V

/0<ji
will be denoted by z^. What precedes proves the following result. T h e o r e m 3 . 1 . Let X be a system with variables w, z and £. Let rj S IN (1 < j < n) be given. Let Pi, P2, • - • ,Pa € k{u>) [ZM] be a set of generators of the ideal of definition ofli(w)^r') over k(w). Then z is observable with respect to w if, and only if, the Jacobian matrix

dP dz(ki)

77(2)

(where 1 < i', < a, and 1 < kj < rj, and 1 < j < n) with a rows and r = Y^j=i(rj + 1) columns is of rank r over \t(w)(z^). R e m a r k 3.2. The determination of the polynomials Pi,--- ,Pa may be not immediate even if the choice of the integers rj £ IN (1 < j < n) is arbitrary as stated by the theorem. These polynomials are the equations of the system X with respect to zM. They result from some elimination procedure, namely, the elimination of £ and (Jjx) yzl

iz2

(h)

(j„)\ 1 • • • izn Jj i > n , - - - ,jn>r„

322

from the original equations of the system (and all their derivatives). To proceed with such an elimination problem we have to differentiate the system original equations some number of times, and then use standard (nondifferential) elimination procedures to eliminate all unwanted derivatives of the variable z. How many times do we have to differentiate the system equations to make sure that we shall get all relations between w and zM by eliminating the higher derivatives of z? This is a longstanding elimination problem still unsolved in full generality. Here is where J. F. Ritt's differential algebraic decision methods come into play. In the next section we show that there is a finite set A consisting of polynomials A\,... , Aat which may replace the polynomials P\,. . . , PCT in the above rank condition, and which are easier to compute. 3.2

The rank condition for rational state systems

In the present section we derive a constructive rank condition for the rational state system (1) with coefficients in a differential field k which, for the sake of simplicity, we assume to be of constants. Let p be the ideal of definition of k(u, y) (x) over k(u, y). Let

Pj(U,X,Y)

= qi{U,X)xf)-pj{U,X)

Qi(U,X,Y)

= fi(U,X)-gi(U,X)Yi

(1 < j < n), (1 < i < p).

The set A consisting of Pi (1 < i < n) is an autoreduced set of k{[/, X, Y} with respect to the ranking

{{Ui,U2, • • • , Um} , { X i , X 2 , . . . ,Xn, Yi,Y2,...

,5^}}-

See Notation 1 for the setup of this notation for rankings. Let Q\ = Qi(U,X, Y) (1 < i < p), and let Qf' be the remainder (see appendix 2 section 2.2.3) of Qy' with respect to A. Q\ is merely the derivative, (Q{°])', of Q{°} in which x\1] (1 < j < n) are eliminated by multiplying n 0)

(<9i Y by Y[qh

and then substituting Pj + pj for qjXf\\

<j<

n). The

(=i

linear combination of Pj (1 < j < n) which appears reduces to zero when the remainder is taken. Explicitly,

323 ( n

\

m

n

nr

at

n

I

n

\

m

a

n.£§^'+E|^n« -* (n.) E ^ 1 ' w

d

9i.Tl„.

i=l

„.VW

' 1=1

If the ground field k were not assumed to be of constants the latter formula would contain more terms. Now this formula is nothing but a counterpart of Lie derivatives yielding the rational expression of yi in terms of y, y, x, u and ii.

We 0

have

Q\ '(u,x,y)

=

0, and Q\ '(u,x,y)

=

0 hence

1)

Q< VA-,l/_))QJ (U,X,»)ep. Let Q\3 ' be the remainder of Q]3' with respect to A. Again, Qi (u, x, y) = 0, hence Q3 (it, X, y) G p. This construction is iterated in order to produce the sequence Q\3' (U, X, Y) such that Q]3' (u, X, y) € p(i
1 < i < p, j G IN): 1% C p.

Let P be a nonzero element of p. Denote by the same symbol P the differential rational fraction obtained by replacing u and y in P by U and Y, respectively. There is S in k { t / , Y } such that S(u,y) jt 0, and SP € k {[/, Y} [X], so that S P G l(X) f] k {U, Y} [X]. As such, P verifies

^SP=

Yl

A P J)

^i

(!)

l
jew +P

for some t' G IN" , and Aitj rewritten in the form

G k { [ / , X , Y } . Clearly, equation (1) may be

PASP= J2 AijPlJ)+ £ l
BitjQ?

1<*
for some t G IN" +P , and BUj (i G IN) are in k {U, Y}[X]. Since the differential ideal of k {U, X, Y} generated by A has no nonzero element in common with

324

I(X) f\k{U, Y}[X] (this results from an obvious degree argument based on the fact that P; (1 < i < n) are with degree 1 in their respective leaders X W ) , the first sum in the previous equality must be zero; showing that P is in (Qi(u,X,y),i 6 IM):/^ We have proved the following result. Lemma 3.3. The ideal of definition of the field k(u, y) (x) over k(w, y) is given by p = (Q\j)(u,X,y), 1 < i < p, j £ 1 N ) : / ^ . We denote

by Q[™\ and

(Q\J)) ^

' jeK

by Q® lor I e IN, I > 1,

(QW) V

/ o<j
and we proceed to show that only finitely many Q\' are needed. Lemma 3.4. Let d ° ( u x y i (1 < i < p), be y i _ i > k<«,a:,i/i,... ,yt) = n\ the transcendence degree oflc(u,x,yi,y2,-• • , j/i) overk(u,x,yi,y2, • • • , J/J_I), p

and let 2_jn'i

=

n

' • ^

e

^ave

n

' — n>

an

^

P = (^i

\uiX,y), 1 < i < p).

The definition of n\ is meaningful since each i/j is differentially algebraic over k(u). Recall that d k(«,y 1 , y 2 ,...,j, i _ 1 ) k ( u ' 2 / 1 ' y 2 '- •• ->Vi) is the the order (minus 1) of the minimal differential polynomial of yi over k(u,yi,y2,. • • ,2/i-i)- Therefore, n

'i ^ d k ( « , y 1 , y 2 , . . . , ! , i _ 1 ) k ( u . 2 / i . 2 / 2 , . - -

,yt).

Moreover, the order, o(X) = d£->k(u,y), of X verifies p

n' < J2 dk<«,»i,»,... ,y i -i) k ( u - 2/1,2/2, • • - , 2/i} = o(X) . Now, from the rationality of x and y over k(u)(x), and from the following fact n

= dk(u)k(u, x, y) = d£
it follows o(X) < n, and the first assertion of the lemma, too. To complete the proof of the lemma, we proceed to show that Q\J)(u,X,y) Let j >n'i, and let Q\J'

G (Q[fyu,X,y):I%

(j > n\).

be the remainder of Q\3' with respect to Pl,P2i

• • • ,Pn,Qi

' •

325

As may be seen from its construction, Q\3' does not involve Y„ if a ^ i or a = i and /? > j , and it may also easily be seen that Q\3' is of degree 1 in Y>3'. Therefore, Q}J' (u,X,y) is free of y derivatives. If it were not the (identically) null polynomial, the fact that Q\J' (U, X, y) — 0 would read as a dependence relation of x over k(w) which would be contradictory. The lemma is thus proved. Remark 3.5. It is usually understood that, when computing the observability rank only the first n — 1 proper derivatives of the output equations need be performed. This lemma provides a proof of this matter of fact, perhaps for the first time. Note that more precisely than stated in this lemma, when computing the observability rank, the ith component of y need be differentiated no more than n\ — \ times. The following lemma is there to save us the need for an explicit knowledge of a basis of p. Lemma 3.6. Let K = k(r) = k ( T I , T 2 , . . . , T „ ) be a finitely generated field extension of k. Let p C k [T] = k [Ti, T 2 , . . . , T„] be the ideal of definition of K over k with generators Pi,P2,... ,Pa. Let a ranking of k [T] be fixed, and let A be an autoreduced set with elements A\,A2,... , Aai £ p, and such that its set, IA , of initials does not meet p. If p = (A): 1^ then

rkK(Pi,P2,... ,P
,Aa.).

For any i G IN, 1 < i < a, there are it 6 IN1"', and a'u € k [T] such that

w = i=iE
i=i

Since r is a generic point of p, and I 4 f| p = 0, we have IAi (T) ^ 0 for all 1 < i < a'. Denoting the value of

2, 01 j

c*f j ( r ) , we have

at r by — , and a - i i ( r ) / / ^ ( r ) by UTi

326

dPi 1

^

dAi

1=1

3

and

This shows that the vectors (dPx

dP2

dP^

. . . . . .

are linear combination over K of the vectors

and vice-versa. Therefore these two sets of vectors have the same rank over K, and the lemma is proved. What just precedes shows the following fundamental result. Theorem 3.7. A rational state system is observable if, and only if, the Jacobian matrix of the vector (Q^](u,x,y)) with respect to x is of rank n over 3.3

h(u,y)(x).

Computing with characteristic sets

The following theorem is a new test of observability based on the construction of a characteristic set of the defining differential ideal of X. Theorem 3.8. Let a ranking ofk{W, Z,T} (where T stands for the differential indeterminate corresponding to £) be fixed, let it be such that any derivative of the components of W is lower than Z\, Z2, • • •, and Zn whose derivatives are all lower that T. Let A be a characteristic set ofl(X). If z is observable with respect to w, then each Zi is (effectively) introduced in A by a differential polynomial of order zero and with degree < d^Zi in Zi (1 < i < n). Conversely, if each Zi is introduced in A by a differential polynomial of order

327

zero and degree d, in Zi then z is observable with respect to w, d^Zj > di, and d^Zi divides di • • • d^ • d\ (hence d°wz\ — d\). Assume that z is observable with respect to w. Let Pi be the polynomial of k{W}[Zi] obtained by substituting W for w in the minimal polynomial of Zi over k(w) and by multiplying by the least common multiple of the denominators of the coefficients in k(W). By multiplying Pi by the product of some powers of the initials and separants of the elements of . 4 Q k { W } , the simultaneous remainders of the coefficients of Pi may be substituted for these coefficients, and then consider Pi as reduced with respect to .4P|k{VF}. Having done that transformation, Pi is still with degree d£,Zi in Zi since the initial of Pi could not have been reduced to zero by the fact that it is not in 1(X) P|k{W}. Now, if Zi is not introduced in A by a differential polynomial of order 0 and degree < d£,zi then Pi would be reduced with respect to A, which would contradict the nonnullity of Pi. Conversely, assume that each Zi is introduced in A by a differential polynomial of order 0 and degree di. It is then clear that z\ is algebraic over k(iy), z-i is algebraic over k(w) (zi), etc., andz„ is algebraic over k(w) ( z i , . . . , z n _i). This implies that z is observable with respect to w. The rest of the proof is classical in the theory of algebraic extensions. By the primality of I(X), and the fact that A is a characteristic set of I(X), the differential polynomials in A which introduce z\,... ,zn, respectively, are irreducible over the fields k(w), k(w) (zi), ..., k(w) ( z i , . . . , z n _ i ) , respectively, hence the degrees over k(w) of k(w) (zi), k(iu) (zi,z 2 ), . . . , k{u;) ( z i , . . . , z n _i) are given by di, dz -d\, • • • ,dn- • • d2-di, respectively. This ends the proof.

3.4

Computing regular observations

Let X be a system with variables w, z and £ such that z is observable with respect to w. Given the discussions in 2.3 we won't be able to compute all the regular observations. Here is a partial result. Theorem 3.9. Let X be a system with variables w, z and £ such that z is observable with respect to w. Let A be a characteristic set of X as in Theorem 3.8. Then an observation which, for each member of A, does not annihilate all its coefficients is regular for the observation of z with respect to w.

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Appendix: Basic differential algebraic geometry The reader may refer to the books 2 ' 9 for more details on the material of this section. Let R be a commutative ring of functions of one variable, with unit, and which contains the field of rationals, Q, as a subring, and let R be equipped with a single derivation, that is, a map of R into R such that (a + b)'=a

+ b,

and

(ab)'=ba

+ ab

for all a, b 6 R where d is the derivative of a. We call R an ordinary differential ring of characteristic zero. All differential rings that will be considered in the sequel will be assumed to be ordinary and of characteristic zero. When R is a field k it is said to be a differential field. The standard fields Q, M and C are differential rings (and fields) of constants. The set, C°°(I, H ) , of real-valued smooth functions defined on a given interval I is a differential ring when equipped with the usual derivation. It is well-known that C°°(I, H) is not an integral domain, hence cannot be embedded in a differential field with the usual addition and multiplication of functions. But the set, C"(1,1R), of real-valued analytic functions is a differential ring and an integral domain. Its quotient field is the differential field of meromorphic functions. The differential polynomial algebra R{T} = R{Ti, T 2 , . . . , TM} is defined to be the polynomial algebra over R in the family of indeterminates (Tj V

)

. The order of an element P G R { T } \ R is

/l
l

JV

denoted by o(P). The object of differential algebraic geometry is the study of the set of solutions (or zeros) of sets of differential polynomials by means of the properties of associated algebraic objects. We shall give to the notion of zero a special meaning here. 1.1

On the admissible trajectories

First we restrict ourselves to differential polynomial equations and inequations (^) with coefficients in a differential field. Next, we look for their solutions (or zeros) in differential field extensions of the ground field of coefficients. These restrictions to fields are for the simplicity of the theory. Proceeding as in algebraic geometry where solutions of polynomial equations and inequations are considered in affine spaces over the algebraically closed field C, we use the notion of differentially algebraically closed field introduced by A. Robinson. A differential field K is said to be closed if it contains zeros of an arbitrary set of differential polynomials with coefficients in K whenever that set of

329 polynomials has a zero in some field extension of K. More precisely, L. Blum provided the following axiomatic definition of differentially algebraically closed fields: for any P,Q € K{T} (here T is single) such that Q ^ 0, o(Q) < o(P), there is r € K such that P(T) = 0 and Q(T) ^ 0. A differential field extension k of k is said to be a differential algebraic closure of k if k may be embedded into any differential field extension of k that is differentially closed. The existence of differential algebraic closures was first proved by L. Blum. As is clear, k is much larger than k, but these two fields have the same cardinality. For more details see 10 . A. Seidenberg 13 ' 14 has shown the following Theorem 1.1. Embedding theorem. Any differential field extension, K = Q ( r i , r 2 ) . . . ,r„), of finite type of the field of rational numbers is isomorphic to a differential field, Q ( r i , T 2 , . . . , T V ) , of meromorphic functions in one variable defined in a domain (open and connected) of the field of complex numbers. This result, a priori, would force us, even for systems with real coefficients, to consider their trajectories with complex time if its real analytic version did not hold. This was obtained by M. F. Singer 15 . Theorem 1.2. Embedding theorem: The real analytic case. Any real differential field extension, K = Q ( T I , T 2 , . . . , T„), of finite type of the field of rationals is isomorphic to a differential field Q ( r i , T 2 , . . . , T V ) , of real meromorphic functions in one variable defined in a neighborhood domain of 0. Of course, as the algebraic closure of the field of reals is not a real field, there is no hope that a differential field possesses a real differential closure. In other words, the Embedding Theorem does not avoid us the need to look at the trajectories of even real systems in abstract differential closures, it rather calls for a decidability result on whether a real system has a real meromorphic trajectories. Such a result was obtained by M. F. Singer 15 : Given a system of differential polynomial equations and inequations with coefficients which are rational functions of the time, the existence of real meromorphic trajectories defined on some open subset of M is decidable. 1.2

Differential algebraic sets

Let fi G IN, [i > 1, let k be a differential field, k a differential closure of k, and let k {T} = k {Ti,T2,... ,TM} be the differential polynomial k-algebra in the differential indeterminates T = (Ti, T 2 , . . . , T^).

330

To any subset £ of k {T} is associated the subset V(S) of kM consisting of the zeros of £ in k , i.e., the elements t = (ti,t2,... ,t^) G kM, such that P(t) = 0 (P G £ ) . It follows V(S) =_V([E]) = V({£}). Conversely, to any subset X of k is associated the subset I(X) of k {T} of elements P such that P(t) =0(teX). 1{X) is a differential ideal of k {T} equal to its radical. A differential algebraic set (ofk ) is a subset X of k such that V(I(X)) = X. The maps, X t-> I(X), and a i-> V(a), between the set of differential algebraic sets of k and the set of the differential ideals of k {T} equal to their radicals, are bijective and inverse to each other. For V I (the composition map of I by V) is the identity map by the definition of differential algebraic sets. The fact that I V is the identity map follows from the Nullstellensatz (see 3 of 12 , and IV.3 of 9)_. A map / : X -» k of a differential algebraic set into k is called a differential polynomial function on X if it is the restriction to X of a differential polynomial map ofk into k, i.e., if there is P in k {T} such that f(t) = P(t) (t G X). The set of differential polynomial functions on X is denoted by k{A"}. k{A'} is easily seen to be a differential k-algebra (for any / G k - ^ } , then / is the differential polynomial function which is the restriction of P to X where P is an element of k {T} whose restriction to X is / ) . Moreover, k {X} is reduced (i.e., with no nonzero nilpotent element) and is isomorphic to k {T}/1 (X) as a differential k-algebra. Let px be the canonical (or natural) surjection k {T} —> k {X} which sends P G k {T} into the differential polynomial function on X defined by P. Clearly, px is a morphism of differential k-algebras, surjective, and with kernel I (X) (the images, through px, of T = (TX,T2,... ,TM) are denoted by T — (TI,T2, . . . , T M ) , and called the differential coordinate functions on X). Hence, as k{T}/I(X), k { ^ } is a reduced differential k-algebra differentially of finite type (i.e., k {X} is differentially algebraically generated by finitely many elements, these elements being, say, r = ( n , T 2 , . . . , T M ) ) . kjA'} is called the differential coordinate ring oi X. To any differential algebraic set X is, thus, associated a reduced differential k-algebra differentially of finite type, k {X}. Conversely, if A is such a differential k-algebra, then A is of the form A = k {T} and the substitution map k {T} —> A which sends P into P(T) is a surjective morphism of differential k-algebras whose kernel, OA, is equal to its radical (since A is reduced). Therefore, to A is associated the differential algebraic set X\ — V ( O A ) (for which Ti,T2,... ,T M are the differential coordinate functions).

331

Roughly, the preceding correspondences, a:X H> k{A^} and /3:A i-> AA, between the class of differential algebraic sets and the class of reduced differential k-algebras differentially of finite type are bijective and reciprocal. For j3a is clearly the identity map, and a/3 is the identity map by the Nullstellensatz. A map / : X —>• y between two differential algebraic sets (X C k , and y C k ) is called a morphism of differential algebraic sets if there are Pi, P 2 , . . . , Pv € k {T} such that f(t) = (P1(t),...,P„(t))

(teX).

It is easily seen that the identity map of a differential algebraic set is a morphism of differential algebraic sets, and that the composition of two morphisms of differential algebraic sets is one (that is, the class of differential algebraic sets together with the morphisms of differential algebraic sets is a category). Now, to a morphism of differential algebraic sets / : X —> y is associated a differential k-algebra morphism / * : k {y} —»• k {X} defined by: /*(a) =

af(aek{y}). The map / i-» /*, thus considered, is clearly injective. Conversely, to a differential k-algebra morphism, :k {y} -» k {X}, is associated a morphism of differential algebraic sets / : X —> y such that = /*. Let yi, yi,... , yv be the differential coordinate functions on y. Let xt = 0(2/0(1 < i < ")• For t G X, let f{t) = (zi(*),... ,x„{t)), we proceed to show that f(t) € y. Let P € I(3>). Then P(yi,y2, • • • ,yv) = 0 in k{3^}, hence (j>(P{yi,y2, • • • ,yv)) = 0 on X. Since P (/(*)) = 0 ( i ' ( y i , » 2 , . . . , ^ ) ) ( t ) = O , we have f(t) 6 ^ ; and the announced assertion is proved. Therefore, the correspondence between the class of differential algebraic sets, and the class of reduced differential k-algebras differentially of finite type, which associates to a differential algebraic set X its differential coordinate ring k {X}, and to a morphism of differential algebraic sets, / , associates /*, is bijective. 1.3

The differential Zariski topology

The empty set and k itself are differential algebraic sets of k . The intersection of an arbitrary family (Xi) of differential algebraic sets is one since

n i ^ = niV(i(-vi)) = v(u i i(^)).

332

The union of two differential algebraic sets X\ and X2 is one again since Xx U X2 = Y{l{Xl)-l{X2)) where l(Xy) -l(X2) is the set of differential polynomials of the form Pj • P2 (Pi £l(X1),P2£l (X?)). Therefore, the set of differential algebraic sets of k is the set of closed subsets of a topology on k called the differential Zariski topology of k**. A differential algebraic set inherits the differential Zariski topology of its ambient affine space. Let a be a differential ideal of k { T } . K is said to be a differential field of definition of a if k • (a PI K{T}) = a; it will also be said that a is defined over K. If K is a differential field of definition of a then every differential field between K and k is a differential field of definition of o. In what follows, a D K{T} is denoted by o K _ Lemma 1.3. Any differential ideal o / k { T } has a differential field of definition. The smallest differential field of definition of a prime differential ideal is differentially of finite type over Q, hence k is a differential closure of it. If b is a differential ideal of k {T} equal to its radical, with differential field of definition K then V(b) = V ( 6 K ) - If a is a differential ideal o/K{T} equal to its radical, then {a}w T i = k-a, (k • o) DK {T} = a, hence {a}jr T \ is defined over K. See 9 , mainly, IV.4 for a proof. This lemma makes clear the following definition. Let X be & differential algebraic set. It will be said that X is a differential K-algebraic set, or is K-closed, or is defined over K, if I (X) is defined over K, which amounts to saying that X = V(I(X)K) = V ( S ) for some subset £ of K { T } . As above, it may be checked that the differential K-algebraic sets of k are the closed sets of a topology on k , which is called the differential Zariski K.-topology. Topological terms which are used relatively to the differential Zariski K-topology will be prefixed by K—, unless K = k. To a differential K-algebraic set are associated a differential ideal of K {T} equal to its radical, and a reducec' c ifferential K-algebra differentially of finite type which is denoted by K {X}. If T\ , r 2 , . . . , r^ are the differential coordinate functions on X, then K{A"} is isomorphic to the differential K-subalgebra of k {X} generated by K and TI,T2,.

.. , rM.

Roughly, the correspondence between the set of differential K-algebraic sets of k , and the set of differential ideals of K{T} equal to their radicals, is bijective, and that the correspondence between the class of differential Kalgebraic s^ts and the class of reduced differential K-algebras differentially of finite type is bijective. A morphism / : X —> y of differential algebraic sets is said to be defined

333

over K if X and y are both defined over K and if / is defined by some differential polynomials with coefficients in K. To such an / corresponds one, and only one, morphism g: K { ^ } -> K{
Differential quasi-affine varieties.

Let k be a differential field, k a differential closure of k. The topological notions used in this section will be relative to the differential Zariski k-topology and we will drop the prefix k-. A differential affine (algebraic) variety is an irreducible differential algebraic set of an affine space over k with respect to the differential Zariski topology. A differential quasi-affine (algebraic) variety is an open subset of a differential affine variety. A differential rational function on a differential quasi-affine variety X is the restriction to X of one on the closure of X. A differential rational morphism of a differential quasi-affine variety X into another differential quasiaffine variety y is defined to be a differential rational morphism of X into

y.

_

For a differential quasi-affine variety X we denote the algebra k {X} by k {X}, and analogously for the fields. A differential quasi-affine variety is said to be defined over a given differential subfield of k if its closure is defined over that field. Lemma 1.4. Let f : X —t y be a dominant differential rational morphism defined over k. Let x be a generic point of X. f^1(f(x)) is finite if and only if[k(X):k(y)] is so. _ Assume that a: is a generic point of X. Since any point in f~1{f(x)) satisfies any equation verified by the generators xi,X2,... , xM of k(X), if the latter field is finite over k(}>) then f~1{f(x)) is finite, too. Conversely, if k(X) is infinite over k(y), then at least one xt is nonalgebraic over k(y). These nonalgebraic components then take infinitely many values, hence, f~1(f(x)) is infinite. The lemma is thus proved. Remark 1J>. According to this lemma, if f~1(f(x)) is finite for some generic point x of X then f~1(f(x')) is finite for any other generic point x' of X. This allows the following definition of generically finite differential rational morphisms. A dominant differential rational morphism f:X—>y is said to be generically finite if f~1(f(x)) is finite for some generic point of x of X. The integer \k(X):]s.(y)] is then called the degree of f.

334

Appendix: Characteristic sets Calculations on a ring usually invoke a basic procedure known as the reduction procedure. In order to provide differential polynomial algebras with such a procedure, it is necessary to define the notion of a differential polynomial reduced with respect to another one as this notion is defined for usual polynomials by means of their degrees. The concept of ranking is the first step towards that goal. Given a differential ideal by means of one of its sets of generators, does there exist some subset of that ideal the computations on which will make easier the answers to questions on the ideal? A characteristic set of a differential ideal aims to play that role. It is not a set of generators of the ideal but it characterizes the ideal, at least when the ideal is prime. The concept of characteristic set goes back to van der Waerden (who called it basic set) and is most known from the work by J. F. Ritt. 2.1

Preorders on differential polynomial algebras

Let R be a nonzero differential ring. Let R{T} = R { T i , . . . ,TM} be the differential polynomial R-algebra in the differential indeterminates T = (2i,...,TM). A ranking of R{T} is a total order on the set of indeterminate derivatives T< N ), consisting of T^j) (j € IN, 1 < i < /x), which satisfies (i) Ti < i f > (1 < i < M ); (ii) T^

< T^

=> Tf + 1 ) < T^'+l)

(1 < i,i' < »,j,j'

G IN).

By means of the bijective correspondence, T( N ) _> I N ; x

IN

the rankings of R{T} are bijectively in correspondence with the total orders on the set 1N£ x IN which satisfy the following conditions: (i) ( i , 0 ) < ( i , l ) (ii) (i,j) < (i',f)

(l<* (i,j + 1) < (i',f + 1)

(1 < *,*' < H,3,j' £ W).

Lemma 2.1 (Kolchin 9 ) . The product order is clearly not a well-order on INt x IN, but it verifies the following property: every sequence of elements of

335

IN* x IN possesses an increasing subsequence whose elements all have the same first component. The lexicographic order on IN* x IN with respect to (j, i) (i.e., the order on IN* x IN induced by the lexicographic order on IN x IN* via the bijective map (*\ J) •-*• UJ) of IN* x IN into IN x IN*) clearly verifies the above properties (i) and (ii), hence corresponds to a ranking of R{T}. R e m a r k 2.2. A ranking o / R { T } is a well-order of the set of indeterminate derivatives T^. This results from the following basic lemma. Lemma 2.3 (Kolchin 9 ) . With respect to any total order on IN* x IN which verifies the above property (i), IN* x IN is well-ordered. A total order on IN* x IN is a well-order if every sequence in IN* x IN has an increasing subsequence. Since, by Lemma 2.1, any sequence of elements of IN* x IN has a subsequence whose elements all have the same first component and which is increasing with respect to the product order, it will suffice to show that such an increasing sequence is also increasing with respect to any order < on IN* x IN which verifies property (i). Let (i,j),(i,j') € IN* x IN be such that (i,j) is less than or equal to (i,f) with respect to the product order, and let e = j ' - j . By property (i), (*> j) < (h j + e)> which proves the lemma. N o t a t i o n 1. We agree in the notation

{{Ti.ieiiMTi.ieM,...} to designate a ranking such that the elements of (Ti)j e /j and all their derivatives are lower than the elements of (Tj)j € j 2 , and the latter and all their derivatives are lower than the elements of the succeeding subset, and so on; in each subset (Tj)^^ the indeterminate derivatives are ranked orderly, i.e., according to the lexicographic order of couples (j, i) corresponding to the indeterminate derivative T^ . It is clear that the rankings of k { T } cannot all be put in the previous form, but this notation will be useful. Let P 6 R{T} - R, and let a ranking of R{T} be given. The leader of P is defined to be the greatest (with respect to the given ranking) indeterminate derivative 7^ which appears in P, and is denoted by up. If dp = d ° p P (the degree of P as a polynomial in up) then P can be put, in a unique way, into the form: dp

P = YJWP i=0

336

where It (0 < i < dp) are in R{T} and are free of up, IdP # 0 and every derivative TJ present in J, is lower than up. The initial of P 6 R{T} - R is denned to be its differential polynomial coefficient IdP in the previous decomposition, and is denoted by Ip. The separant of P is the differential polynomial Y^i=i HiUp-1 (= dP/dup), and is denoted by Sp. It is clear in this context of characteristic zero that the separant of P & R is never the zero polynomial. The set of initials of the elements of a subset V C R{T} will be denoted by I-p, and the set of separants of elements of V will be denoted by S-p, while H-p will stand for the union of Ip and Sp. Leaders, initials and separants are, of course, relative to the particular ranking being used. The preorders that will be considered on R{T} are those which extend the total orders that are defined by rankings. With any differential polynomial P of R{T} — R is associated a couple ui(P) = (up,dp) consisting of its leader and its degree in its leader. By convention, w(P) = (0,0) (P 6 R ) , and 0 is less than any element of T^. The set of couples (u,d) (u = Ooru 6 T ^ ) andd £ IN), is lexicographically ordered. The differential polynomials are ordered according to their associated couples, i.e., it will be written P < Q, and said that P is of lower rank than Q, if u){P) < u(Q). When w(P) = w(Q) it will be said that P and Q are of the same rank. The preorder on R{T} thus defined is of course not an order. Given a differential polynomial P of R{T} - R, IP