Global controllability and stabilization of nonlinear systems

(ZV) V 0 ,

Let D(3?) denote t h e set of all differentiations of K. If Dr,D2e Qfli+)3i}2eD(»)

Z>(5R), then

VQ,/?€R

and

(1.7)

Z>i o D2 - D2 o Dl 6 D ( » ) , where i?i o D2 means t h e composition of two linear operators D\ and D2 ■ T h e set D(jft) endowed with operations (1.7) is called Lie algebra of differentiations of 5R. T h e operation [D1,D2] =

D1oD2-D2oD1

is called t h e Lie bracket of D^ and D2. We call the Lie algebra of differentiation of C°°(M) (C"(M)) Lie algebra of vector fields on M and denote by Lie(p"(M))

(Lie(p"(M))).

Given two vector fields / , b £ Lie(p°°(M)) is also a vector field on M and [Lf,Lb]h

= L[mh

we obtain Lie bracket [/, b] which

(1.8)

\/h£C°°(M).

If

/ fi(x) /i(*) \ /(*) = :

h(x) \ /( hi*) ,b(x)

=

/«(*)I/ V /»(*)

■

{VK(X) M*))/

are representations of the vector fields / , b in local coordinates of a chart (U, ip), where n = d i m M a n d x, = <^(£) (i = 1,2,..., n) for £ 6 E7 C M , then taking in (1.8) /i = ifi we obtain

hm^i

" dh

df,

for i = 1, 2,..., n. Thus in local coordinates of t h e chart (U,
ifM*)=d^]m-

"

dx

^

T h a t generalizes t h e simplified definition of Lie brackets given in Chapter 4 of

Part I.

162 Global controllability analysis Given two C°° manifolds M, N a C 00 mapping g : M —> N induces a conjugate mapping g* : C°°(N) -► C°°(M) (N), where the dot means ordinary real multiplication, g* is a homomorphism of the algebra C°°(N) into the algebra C°°(M). If g : M —> N is a diffeomorphism, then g* is an isomorphism of two algebras C°°{N) and C°°(M). Therefore, for any linear operator L : C°°{N) -» C°°(JV), we can assign the linear operator d'gL defined by g*oLo (ff*)"1 : C°°{M) -> C°°(M). Let the linear operators Lf,Lt, be differentiations of C°°(N). Then g' oLfo

(g*)-1 = Ld'gf

and Ld'g[f,b] =9*° i[/,6] o (5*)"1 = g* o Lf o Lb 0 (g-*)-1 - j ' o l j o Lf{g*Yl L

g* oLso(g*)-

x

og' oLbo{g*y

- g' oLho{g')-^

o{g*)oLfo{g')-

i

=

=

L[d.gftd.gb].

We have proved the following very useful property of Lie brackets of vector fields d*g[f,b]=[d*gf,d-gb], where / , b are C°° vector fields on M. Consider the adjoint operator adf . Lie{p°°(M)) -> Lie(p°°(M)) generated by the vector field / as follows adi(b) = [f,b]. We call adf adjoint operator, because in the literature on Lie algebras the mapping ad : Lie{p°°(M)) -» End(Lie(px(M))), where End(Lie(p°°(M))) is the set of all linear operators from Lie(p°°(M)) to Lie(p°°(M)), is called an adjoint representation. A linear subspace A C Lie(p°°(M)) is called a Lie subalgebra if [A,A]CA,

163

Global controllability

and

stabilization

where

lAA] =

{[f,b];f,beA}.

Given a family of vector fields Fy = {f(x,u); u £ V C R m } we denote by Lie(Fy) the smallest (with respect to set inclusions) Lie subalgebra which contains the family Fy. We say that LieFy is generated by Fy. Consider a flow e":RxM^M, It is easy to verify that [ ( e ^ ) * ] - 1 = ( e - e / ) * and

generated by / £ p°°{M).

| ( e " r = (e"r°£/ = £/°(e"r,

(1.9)

where Lf C°°(M) —> C°°(M) is the corresponding differentiation of the algebra C°°(M). If b £ p°°(M), then making use of (1.9) we derive | { ( e " ) * ° U o [(e")']-1} = ^ { ( e ( / ) * ° i» ° («-")*} = ( e " ) ' oLfoLbo

{t-tfY

- ( e " ) * oLboL,o

(«-«')• = (e<>)* o % 6 ] o [ ( e " ) T \

and hence £(*•(£'/)(> — ■£<J-(e,/)[/,6]

which yields

!( e ")& = )[/,&] or ~(etf)b = d'{etf)adfb. (1.10) at Since (1.10) holds for all b £ p°°(M), we are allowed to introduce the following notation d*{e.t!)b = etad'b. (1.11) T h e notation (1.11) is completely justified when / , b £ p ^ ( M ) . Indeed, if / ( x ) , 6 ( x ) £ p " ( M ) and f(x) = (f1(x),...Jn(x))T,b(x) = (h(x),...,bn{x)f are coordinate representations of / and b, respectively, in local coordinates of a chart (U,(U) there exists a real positive number e > 0 such that the series

d'(e't)b(x) converges to d*(etl)b

= E ( - az£ ) ' ( < 2 V W z ) ) L=o I ,=o '•

for | t |< e. Since

(|)'W)6(*)) l<=o= arf/H^),

164 Global controllability analysis taking into account (1.11) we obtain etad'b(x) =

;=o

Y,adifb(x)1 l -

and, for any x G y>(U) there exists a real e > 0 such that the series converges for all | t |< e. A subset F C Lie(p°°(M)) is called a Lie subalgebra of Lie(p°°(M)), whenever / and 6 belong to F, then [/, 6] also belongs to F. Let Fv(x) = {f(x,u); u 6 V} C p°°{M) be a family of vector fields, where V is an index set. Then we denote by LieFy the smallest (with respect to set inclusions ) subalgebra of Lie(p°°(M)) which contains Fy(x). We say also that LieFv is the Lie algebra generated by Fy. Given a point x € M the integer rank(LiexFv)

=

diia(LieIFy),

where LiexFy is a linear space spanned by {£(1); f G LieFy}, is called rank of the Lie algebra LieFy at the point x G M.

1.2

Groups and monoids.

A set S is called a semi-group if there is denned in S an operation, i.e., a mapping v:S

x S -> S

denoted by v(a, b) = a o b Va, 6 G S,

which satisfies the following condition. Associativity : for each triple a,b,c£ S the relation (a o b) o c = a o (6o c) holds. A semigroup 5 is called a monoid if 5 possesses an identity, i.e., an element e G 5 such that a o e = o Va G S. A monoid becomes a group when every element in the monoid possesses an inverse, i.e., an element a - 1 such that a - 1 o a = e.

165 Global controllability

and

stabilization

E x a m p l e 1.9 Given a C manifold M t h e set Diff(M) of all Cr diffeomorphisms of M is a group where an operating is the composition of two diffeomorphisms. Indeed, if ip,ij> £ Diff(M), then eDiffr(M)

y~l e

Diff(M)

and ideDiff(M), where id is t h e identity mapping. E x a m p l e 1.10 Let Fv = {f(x, u); u e V C R m } be a family of complete C°° vector fields on a manifold M. T h e minimal (with respect to set inclusions ) monoid S(Fv) C Diff°°(M) containing t h e set {e'H")

:

M -+ M ; t e R + , u £ V C R m } ,

where t?Jw is t h e flow generated by t h e vector field f(x,u), monoid generated by t h e family Fv-

is called t h e

Given a group G a set A C G is called a subgroup of G if for any two a, b € A t h e element a - 1 6 is also in A. E x a m p l e 1.11 Let Fv be a family of complete C°° vector fields from Example 1.10. T h e minimal (with respect t o set inclusions ) subgroup of Diff°°(M) which contains t h e monoid S(Fv) is denoted by < Fv > or < f(-,u) >uSv ■ We say t h a t t h e group < / ( ■ , " ) >uev is generated by Fv and call Fy t h e system of generators of < Fv > ■ If B = { E f a i «**.-(*); « = ( u i , . . . , u m ) T G R m } , with {k}f=1 family of complete vector fields, then

C p°°(M), is a

< bj >™=1 or < B > denote t h e group generated by t h e family. If x € M, then < Fv > x ( < B > x) denotes t h e orbit of t h e group < Fv > ( < B >) through x. In this chapter we mostly deal with subgroups of Diff°°(M) (Diff"(M)) and with monoids generated by families of complete C°° or Cu vector fields. Let Q b e a monoid, and let M be a C°° manifold. We say that Q acts on M if there is a mapping H-.Q x M -» M such t h a t n(qop,x) Vq,p£

Q and s € M .

=

tJ-(q,fi(p,x))

166 Global controllability analysis The mapping n{q,x) is denoted simply by qx. The set Qx = {qx; q 6 Q} is called the orbit of the monoid Q through the point x e M. We say that the monoid ( or group ) Q acts transitively on M or Q is transitive if Qx = M \/x e M. Our main objective in this chapter is to give conditions on S(Fv) which ensure that it is transitive.

1.3

Approximative groups and systems on fo­ liations.

Here we introduce the concept of an approximative group and describe the foliation induced by this group. Given a foliation and a flow on a manifold we define a system on the foliation which plays a very important role in the controllability analysis.

1.3.1

Approximative groups

We begin with the abstract definition of an approximative group of a monoid. Definition 1.1 LetX be a topological space and let S be a monoid of mappings of X into X. A group G is called an approximative group of the monoid S if Gx = ~Sx~ VxeX, where Sx is the closure of the orbit Sx. Let M be a C°° manifold, and let Fv = {f(x,u)

6 P°°(M); u G V C R m }

be a family of complete vector fields on M. In this subsection our goal is to construct an approximative group of the monoid S(Fv) generated by the family Fy- We describe three basic methods for the construction of approximative groups: (LC) the method of large coefficients [6,17], (K) Kunita's method [28],

167 Global controllability

and

stabilization

(L) Lobry's m e t h o d [32]. To construct an approximative group we need some mathematical machinery which is based on T h e o r e m 1.1 (Sussmann[42])

Let M be a C°° manifold,

Fv = {f[x, u); u 6 V C R m } C p'+1{M) be a family

and let (I > 0)

of complete vector fields. Then, for any x0 € M, the orbit < Fv > x0

is a connected C

+1

submanifold

of M.

Proof. We define a C ' + 1 structure on < Fv > xQ using the following family of mappings {}-'oe

t f (

"o}U>.

Consider t h e subset II I 0 C TxaM denned by ni0 = where f(v) Diffl+1{M)

span{Uged*gf(v)\X0},

denotes t h e vector field f(x,v) and

with v € V fixed. Since < Fy > C

rf

: -Ll ff ar 0 , ffl»*o(n*o) —

n we have, for all y € < Fv >> Txo,

dim n v = dim II X0 . Let dim II X0 = r, where r is an integer. There is a family of vector fields u; 6 V (i = 1 , 2 , . . . , r ) ,

{d"g1f{v1),d*g2f{v2),...,d'grf(vT)}, which spans I I I 0 , and consequently ranfc{«r f f i/(«i), d'g2f(v2),...,

d~grf(vT)}

= r.

(1.12)

Define the mapping as follows ri

o —

1

1 i

..,yr) = pi" o1eog» « ioe^Mog > Off,.2o... Oft"0g;ooe"^^og ^xo(yuy2,-,yr)=9^oe^Mog x0. B «^W rOft, *l>x<>(yi,y2,' 1

o. .. o ^ o e ' - ' W o M .

It follows from (1.12) that there exists a neighborhood U C R r of the origin such t h a t tpxo :U-^x0isa C ' + 1 immersion of U into < Fv > x0. The family of sets {g ° < M W ) ; g€,W

CU

is open}

168 Global controllability analysis is a base of a topology on < Fv > x 0 ■ The orbit < Fv > xo together with this topology becomes a C' + 1 manifold with the C' + 1 structure generated by the family of coordinate neighborhoods { ( J O ^ ^ ^ - ' O J - 1 ) ; ge,WC

U is an open set}.

Clearly, if go^0(W)ngo^X0(W) for some g,g e < Fv > and W, W C U, then *„ : WD^og-^ogo^iW)

-» W C\{^

og-l)o{gofX0){W)

and ^J" 1 o g_1 o g o ^ I 0 is a C' + 1 mapping. That completes the proof. Q.E.D. It is shown in the proof of Theorem 1.1 that spanllx = Tx(< Fv > x0)

Vx £< Fv > x0.

Hence (d*e'fMf{v)){x)

6 Tx{< Fv > xo)

Vx € < Fv > x0

for all t 6 R and u, v £ V C R m . Since T x (< .FV > x0) is a closed linear space for any x 6 < Fv > x0 fixed, we have Jim i-[(/(<;))x - («re"<">/(»))(*)] £ T x (< Fv > x 0 ) V* 6 R. That yields (d*e t / ' u 'a^ / H /(u))(x) g r , ( < Fv > x0)

Vi € R, x € < Fv > x0,

m

for all u, v e V C R . Having proceeded by induction we come to the conclu­ sion that {ad){u)f{v)){x)

6 Tx{< Fv > x0)

Vx € < Fv > x0

(1.13)

m

for all it, v £ V C R and for any integer j . Therefore the following proposition holds. Corollary 1. Let Fv C p°°(M) be a family of complete vector fields. Then < Lie(Fv) > x = < Fv > x Proof. Since Fv C Lie(Fv),

Vx G M.

169 Global controllability

and

stabilization

it is clear t h a t < Fv > x C < Lie(Fv)

V i e Af.

> x

Let us show t h a t the inclusion < Lie(Fv)

> x C < Fv > x

holds. As Tx(< Fv > x0) is a real linear space it follows from (1.13) that < Liex{Fv)

> C Tx{< Fv > x0)

Vs 6 < Fv > x0,

where Liex{Fv) is a linear space spanned by the vector fields of Lie(Fv) uated at t h e point x. T h e inclusion (1.14) completes the proof. Q.E.D.

(1.14) eval­

We now describe LC-method of the designing of approximative groups. This method can be applied to construct an approximative groups for those families of vector fields which are of the form « = («i,...,Um)T e Rm},

£ ( / , B ) = {/(*) + f > ( * K ;

(1.15)

1=1

where f(x),

&i(x),..., bm(x) G p°°(M)

T h e o r e m 1.2 (LC-method)

are complete vector fields.

The group < bi >T=i

is an approximative group of the monoid 5 ( S ( / , B)) generated by the of vector fields S ( / , B) defined in (1.15).

family

Proof. We use for t h e group < 6,- >?Li

T h a t is compatible with the notation: m

B(x)u

=

Y^bi(x)v.j, i=i

etB" denotes the flow generated by Bu. W h a t we basically need to prove is etBu(x)

C S(f,B)x

Vx e M, u G R m and t £ R.

Given u G R m we obtain ku G R m for any k G R and therefore ef

where

(/+B*«)(a;)

e

5(s(/, S))i

Vfe > 0 and

e *tf+fltl)

Vi G R, i > 0,

(1.16)

170 Global controllability analysis is the flow generated by the vector field f(x) + B{x)u. Since lim eW+Bk»\x)

= etBu(x)

V z e M , f € Rand Vug R m , the inclusion (1.16) is proved. Q.E.D. Corollary 1. The group < Lie(B) > is an approximative group of the monoid S ( £ ( / , B)). Proof. By Corollary 1 of Theorem 1.1 < B > x = < Lie(B) > x

Vx G M.

Since by Theorem 1.2 < B > is an approximative group of £ ( £ ( / , 6)), < Lie(B) > x is also an approximative group of 5 ( S ( / , b)). Q.E.D. A monoid S may have many different approximative groups. For the purpose of the controllability analysis the best we can do is to construct the largest (in the sense of set inclusion) approximative group of the monoid 5. In general it is difficult to find the largest approximative group of the monoid S. It is easier, in some cases to construct any approximative group of S, and then try to enlarge it. We now describe certain simple enlarging procedure proposed in [28]. This procedure is called Kunita's method or K-method. In order to elucidate K-method we next prove the following preliminary results.

Lemma 1.3.1 Let 5R = {£i, ...,£„} be a family of vector fields from p°°(M). Suppose vT(x) G C°°(R x M;TM) is such that vT(x) G p°°(M) for any T G R fixed. Then the existence of a natural number j such that {—YvT{x)eLieJR. dr

VrGR,xGM

and ((—yvT(x))

| r = 0 G Litjt

for 0 < t < j - 1

implies vT(x) G Liejt

VrGR,xGM.

171 Global controllability and stabilization Proof. Without loss of generality we can prove the proposition only for j = 1. Assume -j-uT(aO € L t e , »

VT£R,I£M.

CLT

Let gT{x) denote £vr(x).

Then it follows from

gT(x) G Liex?H V r e R , x 6 i W , that £

lS9rt{x) e Litjt

VT6R,X£M,

where T< = ^ ■ r (i = 0,..., AT — 1). Since lim

N-1

N-* oc i=0

T

J

,.(x) =

"

-[

ge{x)d6

(for a proof see, e.g. [11]), we obtain fT ge(x)d6 6 LieJSt Vr € R, x 6 M. Jo Therefore the integration of gr(x) = Jrf(x) yields vT(x) - [vT{x)] | T=0 G Litjt

Vr G R,x € M,

and it follows from vT(x) | r =o£ £iexSi that iv(x) G Liex9l for all r G R and any x G M. Q.E.D. Lemma 2. Suppose S(Fv) is the monoid generated by Fv = {/(*,«) G p°°(Af); u G V C R m }. Ifrj,^ are complete vector fields such that et£(x) G S(.FV)x, e*"(x) G S(i^v)a: Vx G M and Vt G R+, ifeen

e t(f+,,) (z) e 5(F v )x V x G M a n d Vt G R+.

A proof of this lemma can be found, for instance, in [18]. Let us remark only that the proof is based on the well known fact that weak convergence of controls implies the uniform convergence of trajectories. Let Jft C p°°(M) be a family of complete vector fields. It is convenient to use the following notation: o<4/ = {ad\\adil2...ad,{J;

h + H + - + i„ = J andfc£ «,* = 1,2, ...,i/};

172 Global controllability analysis odd adftf is the subset of ad'^f which comprises all

ad^adg.-.adfj with at least one odd index ik (1 < k < v), i.e., adi£iadij^...adi£vf G odd ad^f if there exists at least one 1 < k< v, such that ik is an odd natural number:

(a4+1/)(^) = {£(*); £ e ad'^f}. Theorem 1.3 (K-method, Kunita [28]) Let Fv = {/(x,u); u G V C R m } and 3J 6e families of complete vector fields from p°°(M). If < 3? > is an approximative group of the monoid S(Fv) generated by Fv and there exist u G V and a natural number I such that (a<4 +1 /(u))(i) C Liex®. Vi G M,

(1.17)

then < Lze(SK U odd adluf(u)) > is an approximative group of S(Fv)Proof. If < 3? > is an approximative group of S(Fv), then by Corollary 1 of Theorem 1.1 < Lie^t > is also an approximative group of S(Fv)Consider vr{x) G C°°(R" x M,TM) which is a complete C°° vector field for any r G R" fixed and is of the form vT{x) = enadh o e 7 ""^ o ... o

T ad e

" ^{f{u))-

£
G Liejt

\/x G M, r € R"

and for all i = (»j, ...,i„) such that ii + ... + »„ = J + 1. Therefore by Lemma 1 u r 0 ) G Lie*5R V i e M and Vr G R".

173 Global controllability and stabilization That yields

E<4i «£.. |i|
T'

V.

(1.18) tn°dtl

0

e ^<.d {2

Q

0

e ^ » ^ ( y ( u ) ) _ „ r ( x ).

Let us denote the left side of (1.18) by Zr. Making use of Lemma 2 we obtain etZr(x) 6 S(Fv){x)

VieR+and

VT € R".

(1.19)

Consider the terms of the highest order from the left side of (1.18), i.e., those terms that correspond to | i |= I. Among them there are such terms that a
£ odd ad^ffa).

We take one of them. Without loss of generality we can assume that the chosen vector field adf1adg...adfj{u) is such that j% is an odd number. Having put r = T+(k), where rt+(k) = 0 for i / j < I ( / i = l , 2 , ^ » ) and T+(k) = k for p = 1,2, ...,*/ we obtain lim

It—<x> K

«£••■,a
■ad£adg. ..adfj(u).

Therefore Hm exp{jyZ^{k)}{x)

= exp{-adi\adg...ad£J(u)}(x)

On the other hand, if we take r = r'(k), rr(fc) = 0 ioii^j,,

where (fi = 1,2,..., v),

T~(k) = k for /x = 2, ...,f and

r-(fc) = - * ,

Vx 6 M.

174 Global controllability

analysis

then ad ■■ ilfW :, *£/(")—

lim

A;—too * ° ° fftc

|.-|
)=

'■

-^adfiadf2...adfj(u). ..adfj(u). ;1 *l 6 T h a t yields

hm e x p ^ ^ - ^ K z ) = eM-^adi[adf2...adif(u)}{x)

Vz £ M.

Making use of the inclusion (1.19) we conclude t h a t < adf1adf2...ad\"J{u) is an approximative group of S(Fv),

>

and hence by Corollary 1 of Theorem 1.1

< Lte(R U odd adlsf(u)) is an approximative group of

>

S{Fy).

Q.E.D. It is important to notice that one can iterate K-method. Indeed, given an approximative group < 3? > of a monoid, by applying K-method for the first time, we obtain the approximative group < £ie(5R U odd adluf{u))

>

Now let 3?o denote SR, and D?i the set iieSRo U odd « S , / ( v o ) , where r0 is the integer equal to /, and v0 = u. By applying K-method for t h e second time we get the approximative group <»2>,

where 5R2 = < £«e3?i U odd ad£ f(vi) natural number.

> for some »i 6 7 C R m and rj being a

Thus a repetitive use of K-method gives us an increasing sequence of approximative groups < 3?0 > C < SRj > C < 3?2 > C ... C < 3?; > C ..., where 5R,-+1 = »,- U odd

adg.f(vi).

If < 3?i > is an approximative group of the monoid S(FV), Fv = { / ( » , » ) e p°°(M);

«£VcE

m

where }

175 Global controllability and stabilization and dim(M) < oo, then spcmx(Zti) C TXM

Vx G M,

where sp<mx(£,-) = span{£(x) G TXM; £ G »,}. Therefore dim(5panI(3?i)) < dim(M), and hence there exists a natural number N such that < fStN >=< Stj > Wj > N. Consider examples illustrating applications of the procedure described above. Example 1.12 Let E(A,I?) be the following family m

E(A,B) = { A x + £>,&;; u = (uuu2,...,um)

G R m },

i=l

where 6, G Rn (i = 1,2, ...,m) and A £ R n x n is a real matrix with n columns and n rows. Clearly Ax, 61,62, ■i6„ are complete vector fields. Hence, by Theorem 1.2 < b, >r=i is an approximative group of the monoid S(£(A, B)). Since ad\Ax = 0 and adsAx = AB, where B = {6i,6 2 ,...,6 m }, AB = {A61,..., A6 m }, having applied Theorem 1.3 we obtain: < AB,B> is an approximative group of S(%2(A, B)). An iteration of K-method gives the approximative group <

A"~lB,A"-2B,...,AB,B>,

where AjB = {Ajbu ..., A J 6 m }. It follows from the well-known fact ( see, e.g.,[39]): n-l

A"B = £

aiA'B

1=0

for some a; G R (i = 0, ...,n - 1), that < A]B, ...,AB,B>=<

An~lB,

...,AB,B>

176 Global controllability

analysis

for all j > n. Thus < An~xB, A"~2B,..., AB,B > is the maximal approximative group of S(Y,{A,B)) which can be constructed out of < B > by K-method. E x a m p l e 1.13 Let us construct an approximative group of the monoid where P

°'*

= {

i

2i+l

l + l xx l ^ ^

S(Polh),

I

I

£

|,-|+j=o

^V;

U

€R}

n

is a family of complete vector fields on R , i.e., x £ R n and a^ e R n for all i and j such that 0 <\ i \ +j < 2k + 1, here i is a multi-index, i.e., i = ( i i , . . . , z„); | z | = E5=i *> a n d a?" = arJ t #J...*j'. By Theorem 1.2 < ao2*:+i > k

is an approximative group of S(Pol ).

Denote by P(x) the polynomial mapping

P(x) = 12 a'°x'|.|=2*+1

We leave for the reader proving the fact that < a02k+i,P(ao2k+i)i

■■■,P'{ao2k+i), ■■■ >,

where P°(x) = x and P'(x) — P(P'~ (x)) for all a; £ R n , is an approximative k group of S(Pol ). The proof of this fact proceeds by induction. However, the general step is completely analogous with the proof of Theorem 1.3 ( for hints see also Exercise 1.6 from Part II). 1

E x a m p l e 1.14 Consider the family of vector fields Y,((, 0 = {( + (u; « € R } , where

e

<-(:)-' -(-L) c=

(*l\ 0

{*»)

,

■'

and f =

/'

\v

It is easy to verify that

KK,d=f ,CI=

° "i

1

~*1

/

( o° \|. 0

■

V\ i ++**ii /

Hence (!<%( ad\(l:== 0,

and since, by Theorem 1.2,

<£>>
<£,adi(> < t, ad(( >

177 Global controllability

and

stabilization

is an approximative group of S ( £ ( £ , £ ) ) .

We now t u r n our attention to the nonlinear systems on compact Riemannian manifolds. Let M b e a C manifold (r > 1, d i m M = n) and TM the tangent bundle of M. Suppose for each x e M the tangent space TX(M) has an inner product

:,w)=

n n

X! 'UiUijffijix),

u,veT3,(M);

•J=l

g*{v,v)>0 (M) e TXx{M) »«(">») > 0 Vu VveT g<x{v, 7 > ^v)) :== 0 only for v = 0, where gx{x) = {g£j{x)}?j=1 is a C " - i matrix function gx : fx{Ux) -* R n x n defined on t h e neighborhood tp\(U\) corresponding to a chart (Ux,
9x,* 9a;,J

vA /

\

for any jr £ fiSJJp D £/*) and x ---

fx13¥V '(!/)•

We say t h a t the set {<7A(x); (Ux, 1) admits a Riemannian structure. Given a Riemannian structure on M we can calculate the volume of a subset D c M a s follows: vol(D) where Cov(D) hoods, t h a t

V\ f u^(D)Ju>nD

J\ gx{xx)

\dxx A dxx A ... A dxx,

= {(Ux,
and

£ = uAt/A n D; dxi A
mf mi

r1 .dxifj) W T H — 3, r / Sfln(r)(

^iePC,I"'(0)=a;,x^(l)=y^

«

dx-'ir)., jar. . . — ,3r— JOT«

178 Global controllability analysis We denote by Br(y) a ball having center y and radius r, i.e., Br(y) = {x E M; p(x,y)
Vi E R,

= vol(A)

(/

where e (A) = {e (x); Vi E A}. Recall that a point x E M is called a Poisson stable point of the flow etf iff, for every neighborhood U of x and for any positive T, there exist £i and t 2 , greater than T, such that ehf{x) E U and

e-' 2 / (x) E U.

L-method is based on the following well-known Theorem 1.4 (Poincare Recurrence Theorem) Let M be a compact Rieman­ nian manifold and f E p°°(M) conservative vector field on M. The Poisson stable points of the flow &*' are dense in M. Proof. Let us denote by P the set of the Poisson stable points of the flow etf. If we prove that vol(P) = vol(M) then the assertion of the theorem easily follows. In other words, we have to show that vol(C) = 0, where C = M \ P. Assume that vol(C) > 0. Then there exist a point x E C, a neighborhood U of x and a positive real number T such that vol(U) > 0, ekT(U) n U + 0 Vfc = ± l , ± 2 , . . . That implies that for all k the system {ehT(U)} of neighborhoods is disjoint, and hence vol{UkekT{U)) =Y,vol(ekT{U)) <1 k

by normalization vol(M) = 1. But the flow etf preserves volume and hence

Zvol(ekT(U)) k

= -£vol(U) k

179 Global controllability

and

stabilization

and t h e last series is divergent if vol(U) consequently vol(C) = 0.

> 0. Therefore vol(U)

= 0, and

Q.E.D. T h e following Theorem gives us Lobry's method or L-method for the construc­ tion of approximative groups. T h e o r e m 1.5 (L-method, Lobry [32]) Let M be a compact Riemannian ifold, and let Fv = {f(x,u); ue V c R m }

man­

be a family of vector fields on M. Suppose there exists a subset W cV C R m such that, for every w £ W, f(x,w) is a conservative vector field on M. Then < f(x,w) is an approximative

group of

>w€w

S(Fv).

P r o o f . We need to prove the inclusion < f(x,w)

>weW

x C S(Fv)x

Vx G M.

(1.20)

If V £ < f(x,w) >wew x, then there exist some real numbers t%,..,,tp and vectors toi,..., wp € W such that y

_ etPfC-r) o e«P-i/(»P-i)

0

... o eh,M{x).

(1.21)

T h e inclusion (1.20) is true if we can find a sequence of positive real numbers {tj,k > 0; j = l f . . . , p } £ i such that yk = etp'kHws') o e ' p - i , ^ K - i )

0

... o

eh'l,Hwi)(x)

and yk —> y, as fc —> oo.

(1.22)

T h e proof proceeds by induction with respect to p. S t e p 1. Assume p = 1, i.e. y = e ( l / ' u ' 1 ' ( x ) . If h > 0, then we put tlik = tx for all jfc. If ti < 0, then making use of Poincare recurrence theorem ( Theorem 1.4) we obtain the sequence of positive real numbers {t\:k > 0}kL1 such that yk = e ' M / C ^ x ) and yk —* y as k —» oo. S t e p 2 . Suppose t h e existence of the sequence of real numbers {tj0;j = l,...,p}kL1 for which (1.22) is true have been shown for p < N. We need to

180 Global controllability analysis prove the existence of such sequence when p = N + 1. Due to the assumption of the induction we have {tjtk > 0; j = 1,.., N}f=1 such that Zl

= eWW0,.,oc''*l(i)

and zj. —» z asfc—► oo, where z

= e'»'W0...oe1''(,"l(i).

By Poincare recurrence theorem, there exist sequence {TJ > O } ^ such that eTj](wN

>(*)">

eljv+i/(<"jv)/^\

as j —> oo, where tf^+i is a real number from (1.21) where p is replaced by JV + 1. We set i/v+i,k = fk for all k and obtain the sequence {tN+i,k,tj,v(k) > 0; j = 1 , . . . , ^ } ^ ! , where i/(fc) is such a natural number that e'w^Wl^j)

G

Bi_(etN^kfiwN)(z))

V/fc > 0,

where Br(y) is the ball having radius r and centre y. It follows that for each V G< f(x,w)

>wGw x

we have yE

S(Fv)x,

while x can be any point of M, and hence the proof is completed. Q.E.D. Here we describe the Zermelo paradox, which is closely related to the Poincare recurrence theorem. Consider a gas enclosed in a finite volume. In classical statistical mechanics a gas is regarded as composed of a large number of molecules, interacting with each other and moving according to the laws of classical mechanics. Thus, such a gas is a hamiltonian system with a very large number of degrees of freedom. It is well known from classical mechanics, that every closed hamiltonian system is conservative. Therefore we are allowed to apply Poincare theorem. Let us take the set of the states of gas in which all molecules are found in the left of the container. Clearly vol(A) > 0. Then it follows from the Poincare theorem, that for almost every x £ A there must exist an arbitrary large number of moments in time when the trajectory started at x e A is in the set A. However, not a single incident has been recorded in all of human history when all the molecules of a

181 Global controllability

and

stabilization

gas returned to occupy half of their container. This paradox is called Zermelo's paradox and is connected with the foundation of statistical mechanics. For its resolution it is commonly said that the cycles of Poincare are so long that they exceed t h e lifespan of the galaxy and, in particular, the lifespan of the gas container. T h e motion governed by conservative vector fields can be very slow. Therefore in applications of L-method to the controllability analysis we have to pay attention to the transient time along trajectories, since it makes no sense to control a system if you need to spend the lifespan of the galaxy in order to reach one state of t h e system from another.

1.3.2

System on foliations

As we shall see in the next section the problem of controllability analysis of a system reduces to constructing the transitive approximative group of the system. However, in some cases the requirement of transitivity of the approx­ imative group is much too restrictive. It can be avoided by resorting to the theory of systems on foliations. For the reader interested in the theory of fo­ liations we recommend the book [45] which contains much of the background material needed for this chapter. A foliation is an extension of the concept of vertical foliation (see Definition 1.5, Chapter I, Part I) and controllable foliation introduced by Definition 2.3 in Chapter II, P a r t I. First, consider n-dimensional Euclidean space R n as R n = R k x R n " k , 0 < k < n, or R n = u (x k+1 ,x k+2 x„)R x (xk+i,x k + 2,...,x n ), that is, R n decomposes into a union of R k x (xk+i,Xk+2, . . . , x n ) ' s each of which is C°° diffeomorphic to R k This decomposition of R n is called a trivial kdimensional foliation or a trivial codimension (n — k) foliation of R n We call R k x (xk+i,Xk+2, ...,x n ) a leaf. Given C manifold M, motivated by this ex­ ample, we call the collection 3 = {Ss*; x 6 M } , where 3 , C M is a C submanifold of M containing the point x 6 M, a C foliation ( with singulari­ ties ) of M if for x ^ y, either Srx = 9 y or S* n SJy = 0. We say that a foliation 9 is a k-dimensional C foliation if 3 satisfies the following. (Fi)

ttx^y,

then either % = % or 5 ^ 9 , , = 0.

(Fii) U i e M 3 * = M. (Fiii) For each x G M, 5$x is an arcwise connected C submanifold of M and dim Qx = dim % for all x, y € M. (Fiv) If d i m ^ = k, then U I g M 3 x is locally C diffeomorphic to trivial kdimensional foliation of R n , where n = d i m M . More precisely, given a

182 Global controllability

analysis

point p G M, there exists a C coordinate neighborhood (U\,
= t / A + i , . . . , x n = «/„},

xk+i

where 2/ifc+i, S/fc+2, ■•■, S/n are coordinates of the point y G U\ in t h e neigh­ borhood V A ( ^ A ) C R n .

?rx is called a leafoi the foliation £r. A k-dimensional C r foliation is also referred as a C codimension n — k foliation. A C chart (U\,ip) which satisfies (Fiv) is called a foliated chart or foliated neighborhood. T h e set of all foliated charts forms a foliated atlas on M. Let FA(S) denote a foliated atlas corresponding to a foliation 3 of codimension q. Then, for ( C A , ^ ) and (V^tp^) G FA(^s) with C/A n U„ ^ 0, t h e m a p W o ^ J 1 : V?A(17A n U„) -> ¥?„([/* n t/„)

decomposes into component functions ( ^ ov?^ 1 )(a; 1 ,X2,...,i„) = (j/i,«/2,--,2/n), which by (Fiv) satisfy 2/i = 2/.(zi,Z2, ••■,£*), i = 1,2, . . . , n - g, (1.23) Vn-q+j = yn-q+j(xn-q+l,Xn-q+2,---,Xn),

j = 1,2,...,?.

We set that a C foliation SJ of codimension q is transversely orientable or coorientable if, in addition to (Fi)-(Fiv), there exist an atlas C J 4 ( 3 ) C J?A(Sf) satisfying the following (CFi) M =

U^^ec^Ux;

(CFii) for {Ux,
>

Q

on the set
183 Global controllability and stabilization It is wellknown (see, e.g. [45]) that a C codimension one foliation 3? of M is transversely orientable or coorientable if and only if there exists a C r - 1 vector field transverse to 3 . Given a foliation 3 of M and W C M which is an open subset of M we say that the foliation 3flW r = { 3 I n l f ; x e M) is a restriction of 3 to W. We call 3 a locally coorientable foliation if, for each 3^ G 9, there exists a. neighborhood W about 3 X such that the restriction of 3: to W is coorientable, i.e., Wc\ 3 is a coorientable foliation. For example, vertical foliations and con­ trollable foliations, in Part I of this book, are coorientable. For the examples of foliations which are not coorientable but locally coorientable, see Chapter IV of the book [45]. Consider a control system of the form T.f : x = f{x,u),

uG V c R m

on a C°° manifold M. Suppose /(x, u) G p°°(M) is a complete vector field for any u g V C R m fixed. The system E^ generates a system on a foliation 3 of M as follows. Definition 1.2 Let 3 be a foliation of a C 00 manifold M and 5 t € S n leaf which passes through x G M. We write = 32

C2f(%)

if -**

=

m

//, for 3 X , 3 Z € 3 , there is a set {9 w }£,i

for some t > 0 and v G V C R such that

Of

c*

-V>(">(z)

_

CV

_ cv

and CS(9= Vj _ l ) = 3fw

fori = 2,3,...,n, then we write 11/(3*) = 3 Z . The correspondence £ / : 3 -> 3 is called u, system on the foliation 3 . The way in which S/ assigns a leaf 3 X G 3 some other leaf 3 Z G 3 is shown in Fig. 1.1. In anticipation of the applications of systems on foliations to the controllability analysis, the following definition introduces the concept of the controllability of a system on foliation.

184 Global controllability

analysis

Figure 1.1: A system S / on a foliation 3 . Definition 1.3 If for any two Q^^y £ 9 we have ~Ef(?sx) = SJy, then we say that the system E / is controllable on the foliation Q. Consider examples illustrating the concepts introduced in this subsection. E x a m p l e 1.15 Consider a trivial two dimensional foliation of R 3 9 = {%;

y € R3}

with 5y = {(xi,x2,x3)

£ R 3 ; x3 = y3}.

This foliation is shown in Fig. 1.2. T h e vector field

(° \ UJ generates the system £ / on the foliation 9 (Fig.1.2). It is easy to see t h a t /(*) =

0

the

system E / is controllable on the foliation $J. E x a m p l e 1.16 Consider a foliation 3 of R 3 generated by t h e leaves each of which is defined as follows % = {x£R3;

|x| = |z|}.

185 Global controllability

and

stabilization

Figure 1.2: T h e system £ j on the foliation 9 , where / = (0,0, x 2 ) 2 The vector field f(x) — x generates the system T,/ on the foliation 3 . We leave for the reader to prove that this system £ / is not controllable on the foliation 3 . However a system £„ on 9 which is generated by any constant vector field a G R 3 is always controllable on 3 . E x a m p l e 1.17 Let < Fv > be a group generated by a family Fv = {f(x,u) e p°°(M); u 6 V C R m } of complete vector fields on a C°° manifold M. Then the foliation 9 = { 9 * = < Fv > x; x e M } is referred as a foliation generated by the group < Fv > It is easy to verify that t h e trivial two dimensional foliation of R 3 from Example 1.15 is generated, for instance, by the group < B > , where

5={

/' * ! \ X2

I>

0/

f-xt\ ,

/M o }. l o ; W * 1

{

.

186 Global controllability analysis

1.4

Necessary a n d sufficient conditions of global controllability.

Here we resort to the theory of approximative groups developed in Section 1.2 in order to investigate the controllability of nonlinear systems. Firstly, we investigate the most simple situation in which a nonlinear system defined on a manifold M possesses an approximative group acting transitively on the man­ ifold M. Secondly, we use systems on foliations to analyze the controllability of those systems that admit approximative groups generating codimension one foliations.

1.4.1

When does the existence of a transitive approx­ imative group imply controllability?

We now investigate the controllability of a system having the form (1.24)

i = f(x,u),

where M is a C°° connected manifold; x £ M,f(x,u) £ p°°(M) and f(x,u) is a complete vector field for any u £ V C R m fixed, where V C R m is a subset in Rm The problem that we are going to attack in this section is to study the controllability of system (1.24) by the following class of admissible controls. PCctm,t(V)

= {«(*) = (ui(t),..., Um{t)); u(t) : R -» V C R m Ui(t) € PCcon,t

and

fori = l , 2 , . . . , m } ,

where PCcnst is defined in Chapter 1, Part I. Recall that the system (1.24) is said to be controllable on M (by if V i . i e M 3T £ R,T > 0 and

PCcon^V))

« ( * ) : [ o , r ] ^ v c R m , u(t) e PCcon8t, such that xu{T,x) == x, where xu(t, x) is a solution of the system (1.24) governed by the control u(t) and satisfies the initial condition X U\T) Xu{t,

X. S) | tt == 00 =: X. x)

In this section our goal is to describe the conditions on f(x,u) and M under which the system (1.24) is controllable on M by the class of admissible controls

PC^iV). The controllability analysis described here proceeds as follows.

187 Global controllability and stabilization We assign the system (1.24) the monoid S(FV) generated by the family of complete vector fields ue VcRm}.

Fv = {f(x,u);

Then we construct a transitive approximative group of S(Fv). We also call this group an approximative group of the system (1.24). If rank(Liex(Fv))

= dim(M) Vi £ M

and there is a transitive approximative group of S(Fy), then the system x= is controllable on M by

f(x,u)

PCconst(V).

Let — Fy denote the family of complete vector fields -Fv

= {-f(x,u);

uEV

cRm}.

The next proposition answers the question which serves as a title of this sub­ section. Proposition 1.4.1 If for any x £ M we have Int(S{-Fv)x)

+ 0,

then transitivity of an approximative group of S(Fv) implies controllability of the system x = f(x,u), u{t) £ PCcontiV), on M. Proof. Let us take any two points x,x £ M and design a control u(t) £ PCcongtiV) which steers the system from x into x. If G is an approximative group of S(Fv), then Gx C S(Fv)x Vi6M and, in particular, Gx C S(Fv)x.

(1.25)

Suppose G is a transitive approximative group of S(Fy). Since Int{S(-Fv)x) + 0, there exists g 6 G such that gx £ Int(S(—Fv)x). Thus, making use of (1.25) one can find a control v{t) £ PCcon^iV) which steers the system (1.24) from x into some point y € Int(S(-Fv)x), i.e.,

188 Global controllability analysis = y

xv(Tv,x)

for some T„ > 0. On the other hand, by the definition of — Fv, we have x €

S(Fv)y.

That yields the existence of a control w(t) g PCCOnat(V), such that xw\i u,, y j

x

for some Tw > 0. Taking

„u wm _ J

«(*)

for

o
~ \ w(t - r.) for r v < t < r» + r.

we obtain X;u(-*u>

+ %,,x)

=

X.

Thus, for any two points i , i £ M, there exist a control u(t) which moves the system from x into x. The proof is completed. Q.E.D. By Proposition 1.3.1, the problem of global controllability analysis reduces to constructing of a transitive approximative group of S(Fv) when =£ 0 Vx € M.

Int(S(-Fv)x)

The methods of checking this condition by computing rank[Liex(Fv)]

for x G M

were considered in [27,43]. Theorem 1.6 ( Krener [27]) Assume M is aC™ manifold and Fv C p°°{M) is as above. If rank[Liex(Fv)] = dim(M), then xG

Int{S(Fv)x)

and xG

Int(S(-Fv)x).

Proof. In theorems of this type we shall only prove one assertion since the proof of the other one is identical. Let U be any open neighborhood of x such that rank{Liez(Fv)) = dim(M) Vz G U. We prove the inclusion x G Int(S(Fv)x) by constructing inductively, for any open neighborhood U of x, a sequence of maps (S(Fv)x)

nU

189 Global controllability and stabilization defined on a sequence of open sets Qj C Rj such that the image tpj(Qj) is a submanifold of dimension j . We continue until j equals dim(M). Choose any u1 G V C R m Let $t > 0 such that e°tH"l1{x) eU

V Sl 6(0,<5,).

We put Q1 = (0, Jj) and V l ( S l ) = e»>j, with y>3{Q') being a submanifold of dimension j < dimM. Choose z G j(QJ) and for all u G V C R"\ /(«,u) G Tz{y3{Q')). This implies that Lie,(Fv) C T,(^-(j{Q')- But this means that r<mfc[£iez(.FV)] < dim(r2(^_,(QJ))) = j < dim(M). Thus, if j < dimM, then by passing to a smaller Q' C RJ we can assume that

/(*, «

i+1

) t z.ivAQ*)) v« 6

9i{QJ)

and also for some 8j+i > 0 e^^"J+1\lpJ(sus2,...,s3))

G !7 V* i+1 G (0,SJ+1).

We define ^• + l(-» 1 ,aj,...,
ea^H"1+1)(
We may choose 6j+i > 0 so small that the Jacobian d

(

\

^ ~ V i + i ( 5 i ! ■■•ishsi+l)

is nonsingular for every (si,.-., s J + i) G Q' x (0,<5J+i) and hence Vj+i

: Q3+1 -

W+1(Q

i+1

) C (5(F v )x) n t/,

where Q-*+1 = Q-' x (0,6, + i), is a difFeomorphism. If n = dim(M), then we obtain Vn : Q " - ¥>„(
{S(Fv)x)n

190 Global controllability analysis for any open neighborhood U about x. And the assertion of this Theorem easily follows. Q.E.D. It is shown in [43] that the property rank(Liex(Fv))

= dimM Vx G M

is generic for C°° nonlinear systems and consequently so is x G Int(S(-Fv)x)

Vx £ M.

Thus, for a generic C°° nonlinear system ( i.e. for almost every nonlinear system ), the problem of controllability analysis reduces to constructing a transitive approximative group of the system. In other words, a generic system is globally controllable if, and only if, it possesses a transitive approximative group.

1.4.2

Full rank conditions of global controllability

Consider a group < Fy > generated by the family of complete vector fields Fv = {/(*,«) G p"(Af);w G V C R m } where M is a connected C°° manifold. There is a very simple sufficient condi­ tion of transitivity of < Fy > on M. Theorem 1.7 (full rank condition of transitivity) The group < Fy > acts transitively on a connected C°° manifold M if rank(Liex(Fv))

= dimM Vx 6 M.

Proof. By Theorem 1.1, the orbit < Fy > x, for each x G M, is a C°° submanifold in M. Following the proof of Theorem 1.1, we obtain dim(M) > dim(< Fv > x) > rank[Liex(Fv)]

Vx G M.

That yields dim(< Fy > x) = dim(M) Vx G M under the condition rank[Liex(Fy)] = dim(M) for all x G M. It follows < Fv > x ~< Fy > y Vx, y G M and < Fy > x = M,

191 Global controllability

and

stabilization

since M is connected. Indeed, if < Fv > x # M , then M = U I g M < Fv > x and there are open sets < Fy > x,

< Fv > y such that

< Fv > x n < Fv > y = 0, where y 6 M\ < Fv > x. This contradicts the fact that M is connected. Q.E.D. T h e full rank condition of transitivity given by Theorem 1.7 becomes a crite­ rion of transitivity when M is a connected C" manifold (i.e., a. real analytic manifold) and Fv Q p"(M) is a, family of real analitic complete vector fields on M. T h e o r e m 1.8 Suppose M and Fv are as above. transitively on M if, and only if, rank[Liex(Fv)]

The group < Fv > acts

= d i m M Vx e M.

Proof. Sufficiency is immediate consequence of Theorem 1.7. We need to show necessity of the assertion of this theorem. Let < Fv > x be the orbit of < Fv > through x G M. It is shown in the proof of Theorem 1.1 t h a t spanlly

= Ty(< Fv > x)

Vi/ £ < Fv > x,

where LTj, = span{Ug£iVevd*gf(v)

|„}.

If < Fv > acts transitively on M , then dim(nB) = dim(M)

Vj/ € M.

Clearly Lzey{Fv)

C n „ = Ty{< Fv > x)

Wy <E< Fv > x.

It remains only to show that Ty{< Fv>x)

= U.yC Liey{Fv)

Vy e < Fv > x.

Since t h e group < Fv > is generated by the set of diffeomorphisms {et/(u); < e R , u e V c R

m

},

192 Global controllability analysis we need only to prove that d*etfM : Lity{Fv)

Vi g R,y g M and u € V C Rm

C Liey{Fv)

Let b g LieFv and b g p"(M). 18, [38]), the vector field

Then, by Cauchy Theorem (see. e.g., section (d*e«Mb)(y)

is analytic with respect to i g R for any y € M fixed. Thus, for any r g R, there exists a real positive number 8 > 0 such that ilZ

(«re"M*)(lO = £

^(d'e^"\ad'f{u)b))(y)

i=0

(1.26)

l

-

for all t satisfying | t — r \< 6. Since, for r = 0, each term of the above series belongs to Liey(Fv), that (d*etfMb)(y) g Liey(Fv)

we have (1.27)

for sufficiency small t. It follows easily from (1.26) that the set of all i such that (1-27) holds is open and closed. That yields (d'etf^b)(y)

g Liey{Fv)

Vi g R.

This is true for any b g Lie(Fv), b g p"(M) and each y g M. Hence we have : Liey(Fv)

-> Liey(Fv)

Vi g R, u g V and Vy g M.

That means T s (< B > x) = n„ C Liev(Fv)

Vy £< B > x.

If < B > acts transitively on M, then T y (< B > x) = dim(M) for all x, y G M. This completes our proof. Q.E.D. Theorem 1.7 and 1.8 together with the results obtained in previous sections of this chapter allow us to formulate various full rank sufficient conditions of global controllability. Historically, most of the first studies on controllability of nonlinear systems have essentially dealt with symmetric systems (see, e.g. [44]), i.e., systems of the form x = f{x,u), u g V C R m ,x g M with the property Fv =

-Fv,

where Fy is the family of vector fields corresponding to the system above.

193 Global controllability

and

stabilization

T h e o r e m 1.9 A symmetric system generated by Fv C p°°{M) is controllable on u connected C°° manifold M if ( and only if, when M is a C" manifold and Fv C p"(M)) Liex(Fv) = dim M V i 6 M. Proof. It follows from Fv = —Fv, that < Fv > = S(FV). Hence the system is controllable if, and only if, < Fv > acts transitively on M. Making use of Theorems 1.7 and 1.8 we complete our proof. Q.E.D. Controllability of nonlinear systems, which are not symmetric, is investigated with the help of the methods of constructing of approximative groups described in Section 1.2 of this chapter. T h e o r e m 1.10 (sufficient

condition of controllability) x =

Let

f{x,u)

be the system generated by G p°°(M); u G V C R m }

Fv = {f(x,u)

which is a family of complete vector fields on a connected C°° manifold M. Suppose 9* C p°°(M) and < SJ > is an approximative group of S(Fv) then the system is controllable on M if = dim(M)

rank[Liex(%)\

Vz e M.

(1.28)

Proof. If (1.28) holds, then by Theorem 1.7 the group < 0 > acts transitively on M. Hence, the assertion of this theorem easily follows from Proposition 1.3.1 and Theorem 1.6. Q.E.D. C o r o l l a r y 1. A system of the form m

£ ( / , B) : x = f(x)

+ £

b,{x)ui,

u = (uu ...,um)T

G Rm,

i=i

where f(x),bi(x), ...,bm(x) G p°°{M) are complete vector fields on a connected C°° manifold M, is controllable on M if rank[Litx{bu

..., bm)] = d i m ( M )

Vx G M.

P r o o f . By Theorem 1.2, < 6,- >™:1 is an approximative group of the system £ ( / , B), and corollary 1 is an immediate consequence of Theorem 1.10. Q.E.D.

194 Global controllability analysis Recall that a subalgebra 7 C Lie(Fv) is called an idealii, for any £ e we have

Lie(Fv),

K,J]ci. We say that an ideal I[B) C Lie(Fv) is generated by a set B of vector fields if 1(B) is the minimal (with respect to set inclusions ) ideal containing B. Corollary 2 Let (f(x),B(x) = {6j(x), ...,6 m (x)}) C p°°(M) 6e complete vec­ tor fields on a connected C°° manifold M. If Vx 6 M

[B,I(B)](x)CspanB(x) and rank(I(B))

= dim(Af) Vx e M,

where [B,I(B)] =

{[t,ri];teB,rieI(B)},

[B,I(B)](x) = {C(x); ( G[B,I(B)]}, I(B)(x) = K(x); C € 7(B)}, £/iera the system £ ( / , B) /rora Corollary 1 is controllable on M. Proof. By Theorem 1.2, < B > is an approximative group of T,(f,B). Ap­ plying Theorem 1.3 we can enlarge < B > and obtain that < 7(5) > is an approximative group of T,(f,B). Application of Theorem 1.10 completes the proof. Q.E.D. Corollary 3 Suppose M = R n ,B = {bu ...,b m } with b{ e Rn (i = 1,2, ...,m) and f(x) is a polynomial vector field of odd degree, i.e.,

f(x) =

Y,

a x3

i >

|j|=2fc+l

where k is a natural number; aj,x 6 Rn for all j and j = (ji,...,jn) multi-index. The system S ( / , B) is controllable on R n if, and only if,

is a

rank(L) = n, where L C Rn is the set of constant vector fields from Lie(f, B). Proof. Necessity. Since 5 ( S ( / , B ) ) C< £ ( / , £ ) >, the controllability of £ ( / , B) implies that the group < £ ( / , B) > acts transitively on R n . According to Theorem 1.8, that means ranfc(iie x (/, B)) = n

V i £ Rn.

Hence rank(L) = rank[Liex(f,B)]\x=0

= n,

Global controllability

and

stabilization

and t h e necessity follows. Sufficiency. By Theorem 1.2 , < &,- >gLj is an approximative group of the sys­ t e m S ( / , B). We enlarge this group by K-method (see Theorem 1.3) and obtain t h a t < L > is also an approximative group of E ( / , B). Thus the sufficiency of the assertion follows from Theorem 1.10. Q.E.D. Corollary 4

Consider the system, ueVcR"

i = f(x,u),

1

on a compact Riemannian C°° manifold M. Suppose there exists a subset W C V C R m such that, for every w £ W, f(x,w) is a conservative vector field on M. The system is controllable on M if rank[LieT{f(w),

w G W}] = dim(M)

Vz e M.

Proof. According to Theorem 1.5 < /(to) >wew is an approximative group of t h e system. Thus this corollary easily follows from Theorem 1.10. Q.E.D. To obtain a necessary condition of global controllability we use the idea of the proof of necessity from Corollary 3. Namely, if a system of the form ueVcRm,xeM

x = f(x, u),

is controllable on a connected C°° manifold M, then the group < Fy > gener­ ated by

Fv = {f(x,u);

ueVcRm}

acts transitively on M. T h e latter gives us the following necessary condition of controllability when M is a real analytic manifold and Fv C p"(M). T h e o r e m 1.11 ( necessary condition of controllability). C" manifold, and let Fv C pu(M). If the system x = f{x, u), is controllable

on M,

Let M be a connected

u g V C Rra, x e M

then

rank[Liex(Fv)]

= dim(M)

Vx G M.

To t h e end of this subsection we consider examples illustrating the applications of t h e results on controllability obtained above.

E x a m p l e 1.18 Consider a linear system having the form m

S(A, B) : i = Ax + J2 «»&«>

196 Global controllability analysis where b{,x 6 Rn (i = l,2,...,m) and A G R n x n By Corollary 3 of Theorem 1.10, the system E(A, B) is controllable if, and only if, rcmk{B,AB,...,An-1B}

= n.

Example 1.19 Let us investigate controllability of the system 2

2k+l

!+ I - ^

yZ o-ijx'u3

|.|+i=o

where u £ R and ciij, x e Rn for all i, j . In Example 1.13 it was shown that < O.02k+l,P(a0

2k+l)i ■■■!-F"(a0 2A+l), ■•• >

is an approximative group of the system. By Theorem 1.10, if = n,

rank{a02k+1,P(a02h+1),...,P'{a02k+1),...}

(1.29)

then the system is controllable on R" We can choose the parametrization dr dt

1 1+ | x(t) |2*+i

of all the trajectories of the system. Then dx ~T~=

2k+1

5 J aijx'u3

(1.30)

and this polynomial system is controllable on R n if, and only if, the origi­ nal system is controllable. Thus, it follows the polynomial system (1.30) is controllable on Rn if (1.29) holds. Example 1.20 On the sphere S 2 = {x € R3; | x |= 1}, consider the system x = u1bl(x) + u2b2(x), u{ = {0,1} (»= 1,2) defined by the two vector fields bi(x) and b2(x) whose trajectories are "longi­ tude" defined for two different axes. For example / X2 h(x) = ( —X! \ and b (x) = 2

I 0 }

0 X3

V-x

2

1 )

These two vector fields are analytic and conservative. Corollary 4 of Theorem 1.10 implies controllability of this system on S2

197 Global controllability

and

stabilization

E x a m p l e 1.21 Consider the system '

d1 dt

X j \ Z2

V x3

/

0

(

\ Zl

\ \+Ul\

|'

1

\

* Xj

V

\

Xl

( - * J

+ u2

0 >1

v

\

II

+ U3

V o ,/

1f l \ 0

\V

0 /

Using LC-method ( Theorem 1.2) we obtain that < B > , where B is defined in Example 1.17, is an approximative group of the system. K-method gives us the approximative group

(o\ <

0

,B>,

\i and therefore, by Theorem 1.10, the system in question is controllable on R 3 .

1.4.3

Systems on foliations and controllability of non­ linear systems.

We have seen in the previous section that the problem of controllability analysis of a generic system reduces to constructing the transitive approximative group of the system. For some types of systems, this problem is solved by Theorem 1.10 and its corollaries. However, in some cases the requirement of transitivity of the approximative group is much too restrictive. It can be avoided by resorting to the theory of systems on foliations from subsection 1.3.2 of this Chapter. Consider t h e system x = f(x,u), ueV cRm, (1.31) generated by the family, Fv = { / ( a , u ) £ P°°{M);

Rm},

ueVC

of complete vector fields evaluated on a connected C°° manifold M. T h e o r e m 1.12 Let < B > be an approximative Suppose Qx e Int(S(-Fv)x) Vx 6 M and

group of the system

(LSI).

Q = {5X =< B > x; x 6 M} is a foliation of M. Then the system (LSI) is controllable on M if, and only if, the system Y,f generated by f(x, u) is controllable on the foliation $s. T h e assertion of this Theorem easily follows from the concept of a system on foliation, see Definition 1.2. To check details is left for the reader as an exercise.

198 Global controllability analysis Thus, solution of the controllability analysis problem reduces to investigation of controllability of T,f on 9 induced by an approximative group of the system (1.31). We will give sufficient conditions and criteria of controllability of systems on a foliation 9 of codimension one when the foliation is locally coorientable, or locally transversely orientable, i.e., every leaf 9 , ^ 9 has a neighborhood 0(QX), in which the restriction of the foliation 3* is coorientable in the usual sense. A formal definition is given in Subsection 1.3.2. We introduce the set £ / C 9 of leaves with the following properties: 9 X e Ef if, and only if, there exist points y, z G 9 X and w, v G R m such that the coorientation defined by the vector f(y,w) is opposite to the coorientation defined by the vector / ( z , v). More precisely, let S be a codimension one foliation of M, and let CA(S |o(3*)) be an atlas compatible with some coorientation on a neighborhood 0($sx) about 3 * (see Subsection 1.3.2). Then, for any two charts (Ux,
{

Vn =

Vn(xn)

dyn(xn) ^ Q dx„ on the set
(U,0,(#,0)eCA(S>|O(o.)) such that y£U,zeU and fn{y,w) • f„(z,v) < 0, where f(y, w), f(z, v) are coordinate representations of the vector fields f(x, w) and f(x,v) in the charts (U,tp) and (U,ip), respectively. The case M = R 3 and 9* = {x3 = const} is shown in Figure 1.3. Given a foliation 9 of a manifold M, let us denote by P the mapping P :M - . 9 , P(x) = 3 , . The mapping P : M —> S is called projection. Consider the topological space (SJ,T) with topology r defined as follows: A 6 T if, and only if, P~X{A) an open subset in M.

199

Global controllability and stabilization

Figure 1.3: Q'x £ Ef, because the vectors f(y, w), f(z, v) induce different coorientations about 9 L . We denote ( 2 , T ) by M / 2 \ Owing to this, the topology r is called quotient topology. The following theorem gives a sufficient condition for controllability of S/ on 3. Theorem 1.13 Suppose M'isa connected manifold. If a foliation 9 of codi­ mension one is locally coorientable, then Ef = $s implies the controllability of Sy on 9 . Proof. It easily follows from Ef = 3 , that, for any 5$x £ 9 , the set {%; E , ( 9 . ) = %} is closed and open in quotient topology. Therefore

9 = {%; Zf(%) = %} for each 9 X € 3 . This completes the proof. Q.E.D. In some cases, the equality Ef = £r provides also a necessary condition for controllability of £ / on 3 . Theorem 1.14 Suppose M is a connected manifold. Given a locally coori­ entable foliation 9 of codimension one, such that for every x £ M the set M\QX is not connected, then T,j is controllable on 9? if, and only if, Ef = $s.

200 Global controllability analysis Proof. Sufficiency follows from Theorem 1.13. To prove the necessity we notice that E} ^ S yields the existence of 3 , £ 9 such that 3 X ^ Ej. Since M \ 3 ^ is not connected, there exists a leaf % 6 3 such that £/(5») / %■ That means the system Sy is not controllable on 3 when Ej ^ S. Q.E.D. The condition Ej = 9 is fairly burdensome. It may be weakened if we intro­ duce the class of pre-Hausdorff foliations. A foliation S o f a manifold M is called pre-Hausdorff if M/$s is homeomorphic to a gluing of at most countable many intervals and half-intervals from R by homeomorphisms of their open subsets. For example, consider the interval I = {x G R; 0 < x < 1} and half-interval J =

{y€R;0
Suppose g : A —» g{A) is a homeomorphism of some open subsets A C I,g(A) C J. Introduce the equivalence relation " ~ " on the set I U J as follows: x ~ x if either i £ / \ A o n 6 J \ 9{A)\ x ~ g(x) when x G A, and consequently 3_1(2/) ~V for y G g(A). Let $sx be the equivalence class of x, i.e.,

%= {yelu

J;

y~x}.

We call the quotient family (7U J)/SJ endowed with quotient topology a gluing of I and J by the homeomorphism g of the open sets A, g(A) and denote (I U J ) / S by I U3(yi) J. An embedding of I Ug(A) J into R2 is shown in Figure 1.4, when g = id and A = {x G R; | < x < | } . It is easily seen that the gluing is not a Hausdorff topological space. For instance, the points of I Ug(A) J (see Fig. 1.4 ) corresponding t o i = \, x = § from 7 and J are not Hausdorff separable. Considering a pre-Hausdorff foliation 9 we can assign a gluing M / 9 Hausdorff space if we identify the points of M/^s that are not Hausdorff separable from one another. This Hausdorff space is called the gluing graph of M/S?. The gluing graph of I U3(A) J is shown in Fig. 1.5. A system £/ on a pre-Hausdorff foliation 3 induces the orientation of the gluing graph of M / 9 and augments its vertex set with new points. More precisely, let {Oi}| =1 be the set of arcs of the gluing graph of M / S , and let us fix any orientation on each of the arcs. We define a quasi-ordering of M / 9 as follows:

201 Global controllability

and

stabilization

Figure 1.4: / ug(4) 3 is the gluing of I and J by the homeomorphism g = id of A = {x € R; | < x < | } and #(^4).

Figure 1.5: T h e gluing graph of / U S (A) J-

202 Global controllability

analysis

$5X -< 3 y if E / ( 3 v ) = 3 , , and we move from 3 ^ into 3?B along the tra­ jectories of the system x = f{x,u) in accordance with t h e orientation of the gluing graph of M / 3 . Consider the sets E+(M/3),E-{M/5). We set % G E+{M/%) ( 3 , G E~(M/^)) if there exist r > 0 , * G $ „ and « G PCconst, u :[0,T]-+V such that for any fj < t2, h,t2 G [ 0 , T ] we have 3wj(ti,*) -^ ^ j ( ( 2 l J ) (3*}(i»,*) -< 3«y(«i,«■)). where a;*(f,z) is the solution for the initial value problem i(t)

=

x(0)

=

f(x(t),u(t)), z.

We add the end points and connected components of t h e sets P[E+(M/$S)),

P(E-{M/$))

to the vertex set and the arc set of the gluing graph of M/Q*, respectively. T h e digraph obtained is called a digraph of S / on the foliation 3 and denoted by T/(^s).

Definition 1.4 LetTfC^) be a digraph ofHf on a foliation 5 . r^(Sf) is said to be transitively connected if it is strongly connected and ifEf(^sx) = ^sy for any two points P ( 5 I ) , P ( 3 y ) G IV(3) such that there is a directed path which can be used to reach P ( 3 y ) from P($?x) ( of course we have to go along the path in accordance with its orientation induced by the orientation ofTf(^s) ) . The following proposition is an immediate consequence of t h e introduced con­ cepts. T h e o r e m 1.15 Let 3 be a pre-Hausdorff foliation. and only if, r / ( 3 ) is transitively connected.

S / is controllable

on 3 if,

Corollary 1 If M/Q is homeomorphic to S1 = {x G R 2 , | x | = 1}, then S / is controllable on M if, and only if, either E+(M/S) = M / 3 or E~(M/$s) = Af/9. We recommend the reader as an easy exercise to prove Theorem 4.3 from Chapter 4 of Part I using Corollary 1 of Theorem 1.15. E x a m p l e 1.22 Let us investigate controllability of t h e system

d 1f * t \ dt \ K*3 I

I''

v.,H \

(

X2 -Xi

\

I

M

0

N

\ V.2

-S3 .

x

2

,

203 Global controllability and stabilization on R3. By LC-method, < B >, where B=

1 X2 \{ U o;

° \\

-X3

v *. /

is an approximative group of the system. The foliation {< B > x}IgR3 co­ incides with the foliation 9 from Example 1.16. The system on foliation £ / generated by

('] f{x) =

0

w

is controllable on 3 . Hence it follows from Theorem 1.12, that the original system is controllable on R3.

1.5

Hypersurface systems.

In this section we investigate the controllability of a hypersurface system de­ fined as follows. Consider a nonlinear system of the form x = f(x,u) u e V c R m ,

(1.32)

where x 6 M; M is a connected C°° manifold and Fy — {/(*>«) 6 p°°(M); u € V C R m } is a family of complete vector fields on M. This system is called a hypersurface system if it possesses an approximative group G such that there exists a C 1 submersion W(x) : M - . R , and Gy = {x 6 M; W{x) = W(y)}

(1.33)

for ally e M. In analogy with the theory developed in Chapter II of Part I we call 9 = {%}yeM, where (1.34) % = {xe M, W(x) = W{y)}, a controllable foliation generated by the approximative group G of hypersurface system (1.32). To define an analog of equilibrium set i^_1(0) from Chapter II of Part I we introduce the function $(x,u) = [-W(e^")(x))]| t = 0 ,

(1.35)

204 Global controllability analysis $ : M x V -> R, where W(x) is the C1 submersion from (1.33). The function $(x,u) plays the same role in the controllability analysis of hypersurface systems as the function f{x) in the controllability analysis of cart garlands ( see Chapter II, Part I). Definition 1.5 Equilibrium set of a hypersurface system (1.32), which has a controllable foliation (1.34), is defined to be $ _ 1 (0) C M x V where $-!(()) = {(x,u) eMxV;

*(»,«) = 0}.

The next theorem follows directly from Theorems 1.12,1.13. Theorem 1.16 Suppose f(x,u),W(x)

and 9* = {^SX}XSM

x 6 IntS(-Fv)x

are as above. If

VieM

and for each 3 , E 3 there exist y,z £ SJX and v, w 6 V such that $(y,v)$(z,w)

< 0,

then the hypersurface system is controllable on M. Theorem 1.14 together with Theorem 1.16 imply the following proposition. Theorem 1.17 Suppose all conditions of Theorem 1.16 are fulfilled. More­ over, assume M\^sx is not connected for any ?sx £ S. Then the hypersurface system (1.32) is controllable on M if, and only if, for any 9 , £ 9 there exist y,z £ Ssx and v,w £ V such that $(y,v)$(z,w)

< 0.

Let 9 be a foliation of M. We denote by 9 x V the foliation

3= x V = {% x V}X^M of the manifold M x V. Having fixed u £ V we obtain the function «(-,«) : M - > R , where $(x,u) is defined in (1.35). If, for any u £ V, this is a C 1 function of x, then, for any b £ p°°(M), Lb$(x,u)

205 Global controllability

and

stabilization

is correctly defined. C o r o l l a r y 1 Let x = f(x,u),

cRm,

ueV

be u, hypersurface system which has a controllable foliation 3 generated by an approximative group < B > of the system, where B = {bi(x),..., k(x)} C p°°(M) is a family of complete vector fields on a connected C°° manifold M. If all conditions of Theorem 1.16 are satisfied, i

J2{Lbi$(x,u))2

14-1(0)/ 0

>=i

and then the hypersurface

(Q^x V)n$- 1 (o)/0 VxeM, system

(1.32) is controllable on M.

Corollary 2 Suppose all conditions of Theorem 1.17 are satisfied and the hypersurface system (1.32) has a controllable foliation 3 generated by an ap­ proximative group < B > of the system where B is defined in Corollary 1.

If l

E(^*(x, «)) 2 1.-■»(0)^ t=l then the hypersurface

system

°i

(1.32) is controllable on M if, and only if,

(3« x F ) f i r ' ( 0 ) / «

VxeM.

So far we have assumed t h a t a manifold M has no boundary, dM = 0. We now t u r n our attention to the controllability analysis of hypersurface systems on a connected submanifold D C M such that dimD = dimM,

dD and 3D is a C°° submanifold of M,

+%

Aim{dD) = d i m ( M ) - 1.

The following theorem gives us a necessary condition of controllability of a hypersurface system on D C M. T h e o r e m 1.18 Let M be a connected C°° manifold, and D C M be a con­ nected C°° submanifold defined above. If all conditions of Corollary 2, where M is replaced by D , are met and the hypersurface system (1.32) is controllable on D, then sup W = sup W, D (Ox^no-ifo)

206 Global controllability

analysis (1.36)

inf W = inf W, D (DxVJn*- 1 ^) where the submersion

W(x)

is defined in

(1.S3).

Proof. It is clear that sup W >

sup

W

and infVF< inf W. D (DxV)n*-i(o) We have to show that the opposite inequalities also hold. If sup W > sup W D (DxV)n»-'(o) or infW< inf W, D (DxV)nt-'(o) then there exists x 6 D such that 3 , n ( i ) x F n * _ 1 ( 0 ) ) = 0- Hence the hypersurface system (1.32) is not controllable. T h e contradiction obtained completes the proof. Q.E.D. For some hypersurface systems, the conditions (1.36) becomes a criterion of controllability. Consider, for example, a linear n-dimensional system of the form n-l

T,(Ax,B):

i = Ax + ^2 km, i=i

where x € R",6i (see Theorem 1.2 generated by B = generated by < B

€ R" (» = ), the system {b\,..., fe„_i}. > consists of WB(x)

l , . . . , n - 1) and A G R " x " . By LC-method T,{Ax,B) has an approximative group < B > It is easy to see t h a t t h e controllable foliation level surfaces of t h e function = det(x,6i,...,6n_i),

where det(x,6j,...,6„_i) denotes the determinant of t h e m a t r i x composed by columns (x,&i, ...,6„_i). In this case the function $ ( x , u ) defined in (1.35) becomes $A,B(X) = det(Ax,b1,...,bn_i). Recall t h a t a domain D C R n is called convex if Vx,2/GD

TX + (1 - r)y € D for all r € [0,1].

207 Global controllability and sta.biliza.tion T h e o r e m 1.19 IfD C R2 is a convex domain, with Int(D) jt 0, then E(Ax, 5 ) is controllable on D if, and only if,

*lffl(O)n/nt(2>)^0, ronfc{A5,B} = n and sup WB = D

sup Dn

WB,

*A' B (o)

(1.37) infWB= D

Dn

inf WB. *x^(°)

Proof. The proof is completely analogous with that of Theorem 1.4 from Chapter l,Part I. Necessity easily follows from Theorem 1.18. In order to prove sufficiency we have to show that y e D implies Qy n (D n $ ^ ( 0 ) ) / 0, where % = {x e D; WB(x) = WB(y)}. Indeed, by Corollary 2 of Theorems 1.16,1.17, that means controllability of S(Ax, B) on D. Let us take any y e D. Then inf WB < WB(y) < sup WB, D D

and hence it follows from (1.37) that inf

WB < WB(y) <

D^A'.BW

sup

WB.

Dn*-^(o)

Since WB(x) is a linear function and D is convex, there exists a point z G D n $^ B (0) such that WB(z) = WB{y). This implies 3 , n ( D f l *x,s(0)) 7^ 0 and the assertion of Theorem 1.19 easily follows. Q.E.D. In conclusion of this section we consider several examples of the controllability analysis of hypersurface systems. Example 1.23 Let us investigate controllability of the system S(Ax,S) on the convex domain D = {XGR";

<X,TX><1},

where rank(B) = n — 1; < x,Tx > is a positive difinite quadratic form, i.e., < x, Tx » 0 for x G R", x ^ 0. In order to find a maximum and a minimum of the function WB(x) on 3D we construct the Lagrange function I ( x , A) = WB(x) - A(< x, Tx > - 1 ) .

208 Global controllability analysis It is well known (see, e.g., [11]) that WB(x) can have the maximum or the minimum at the point x* only when dL{x*,\')_^ dx dL(x*,\') d\

0

for some A* ^ 0. Therefore = 0,

p-2\*Tx*

<x*,Tx* > = 1, n

where p e R is such a vector that = 0 Vi = 1,2,..., n — 1 and WB(p) =| p | 2 We denote p by bi x b2 X ... X 6„_i and call a vector product of b\, b2, ...bn-\. Thus we obtain that the function WB(x) attains the maximum at the point

„

■L+ -

I V
rp-

-i„

V>

fi

and the minimum at x*_

1

—T-

V>

v
V,

where p = b^xb2x ... x fe„_i. Hence Ti(Ax, B) is controllable on D if, and only if, rank(AB, B) = n and I ; , I * _ £ f _1 (0) CiD = {xeD;

det(Ax, 6 lf ..., 6„_i) = 0}

or, in other words, =0 and rank(AB,B)

= n.

Example 1.24 Consider the system , / Xi \

1 X\ + X% ^

1

4::i \ ,

| +Ui

(°N0

i

\+u2\

\ 1 / 1

0 1 \vO

/'

where a; = (xi,x 2 ,X3) T £ R" and « = (ut,U2)T € R2. For this system <&(x,u) = —x2 — x\. The controllable foliation is defined by planes parallel to the plane

209 Global controllability

and

stabilization

x\ = 0. Theorem 1.17 directly implies that this system is not controllable on R3 E x a m p l e 1.25 Consider the system

d 1f dt

1 x2 + x3 \

X! \

«. ,K*3

-[ V ;i

)

+ «!

i^ 0 *1 + 0

'V 1 /'

,/

1r o \

H \ 0i / i

In this case $ ( i , u ) = —xf — x\ and the controllable foliation 3 is defined by the planes X\ = const. On every leaf 9fx there are points y,z G^sx such that -2/1 - vl < 0 and - z\ - z\ > 0, therefore Theorem 1.16 implies controllability of the system. E x a m p l e 1.26 Let us investigate controllability of the system i

=

1 TT\—IM+T

2k+1

^

aijx'u3,

(1.38)

where u G R and a,-.,-,x G R" for all j and i = (ii,...,in). was shown t h a t < 0 0 2t+l,-P(ao2A+l), ■ ■,-P(ao2fc+i),-

In Example 1.13 it

.. >

(1.39)

is an approximative group of the system, where

P(x) = 5Z a'°x'|i'|=2fc+l

Let rank{a02k+i,P{ao2k+i),---,P{o-o2k+i),---}

= n - 1,

1

and {C.}"^ the basis of the linear span generated by the vectors {<20 2*:+l,-P(a0 2ifc+l),---,-P( a 0 2*:+l),---},

then the controllable foliation 9 generated by the approximative group (1.39) is as follows:

3 = {QyJseR", 3» = { i 6 R " ;

det(a;,C 1 ,..,C„-i) = det(3/,Ci,...,Cn-i)}.

By Theorem 1.17, the system (1.38) is controllable if, and only if, for an arbitrary c G R there exist y+, y- G R n and u+, u_ G R such t h a t d e t ( y ± , C i , - , C n - i ) = c,

210 Global controllability analysis and we have 2fc+l

±det( J ] oyir±«i,Ci»—)Cn-i) > 0l«'l+i=o

This proposition includes a larger class of systems of type (1.38) than the full rank condition obtained in Example 1.19. For example, the system i i = u, x2 = x 3 , x 3 = u3 is controllable on R3, while the full rank condition from Example 1.19 is not satisfied.

1.6

Exercises

1.1 Prove that a metric topological space is Hausdorff and an infinite set with Zariski topology is not Hausdorff. 1.2 Given a connected topological space X and A C X, show that if A is simultaneously closed and open in the topology of X then A = X. 1.3 Let RP" denote the real projective space of lines in R n + 1 , i.e., RP" is quotient space R n + 1 / ~ , where x ~ y if there exists A £ R, A / 0 , such that x = \y. Prove that RP" is a C" manifold and dim(i?P n ) = n. 1.4 Consider a monoid generated by two complex numbers e^*' and 1. Show that the group 5 1 = {e1*; ( 6 R} is an approximative group of this monoid. 1.5 Let M be a C°° manifold and / , b 6 p°°(M). Prove that C™*(Q / ) =

(eTfc)»e™">(/),

where a : M -> R is a C°° function, and (e r6 )*(a)(x) = a(erb{x)) for all x e M. 1.6 Consider the system

ITT^rPW'

x e Rn

'

where P(*) =

E |i|=2t+l

«««>«'• a « G R n Vi = (it, ...,i n ).

211 Global controllability and stabilization Prove that

^orLi^,(P)^)-p(b) for any constant vector field b e Rn. Hint: use

a

... + 6„A ) .p

+

and the well known fact that

<-<'^,f4

where | j \= ji + ... + j„, j = Q't, ...,i„), p. = / J L . J „ ! and. 53'

aiii jl

dx->

dx

...dxj»'

1.7 Consider the mapping i/>(t) = e _ t t j 0 e-"" 0 e"* 0 e(i,1(x0), where x 0 € M, M is a C°° manifold and bi,b2€ p°°(Af). Prove that for x = ip(t) we have -b2(x)-(et"d'*b1)(x)+(et°d'ioet°dhb2)(x)+{et'"'>* oe""^o«^*"*>S, ) ( x ) V t € R .

^V(*) = at

1.8 Using Exercise 1.7 show that V>(i) = x 0 + adh b2(x0)t2 + 0(t3) as t -► 0. 1.9 Prove that the system d If * l \ x Jt \ 2 K*3 J

is controllable on R

1

M \

-{\' x

V 2 ,

1

x3 '\ )

+

v 0 ,

i * |

|f

l

\

; ))

3

1.10 Show that the system

d( Xl x \ = (

— dt

\x3)

2

Xl x2

)

\x3 + lj

is not controllable on R 3

+ uj

1 -^* t \

\ 0 )

(

+ uB

0

)

{- -x2 )

212 Global controllability analysis 1.11 Prove that the system Xl

X2

x3

= "1, = ua, = X\U2

— X2U\

is controllable on R31.12 Show that the system

= = *3 = £4 = is = is controllable on R5.

ij

«1,

X2

«2,

XiU 2 — X 2 Ul I1U2 2 -Z2"l

213

Chapter 2 Local stabilization of nonlinear systems. In this chapter we investigate local stabilization of nonlinear systems. Here we consider only time-independent (stationary) state feedback of the form u = u{x). This assumption allows us to formulate rather complete results on local stabilization of nonlinear systems. On the other hand, there are known only few results on local stabilization by a time-dependent state feedback, u = u(t,x). The chapter is organized as follows. In the first section we give a precise mathematical statement of the problem and formulate the simplest necessary and sufficient conditions of local stabilization, the second section is devoted to the local stabilization analysis methods based on Liapunov's direct method. In the third section we use some modifications of the center manifold approach in order to design smooth local stabilizers for nonlinear control systems in critical cases.

2.1

The simplest necessary and sufficient con­ ditions of local stabilization

In this chapter we consider the problem of local stabilization of control systems of the form

S/.

x = f(x,u)

n

where states x(i) evolve on R , controls u(t) take values on R m , for some integers n and m. Our main interest is in finding feedback laws u = u(x),

u(x*) = u*,

214 Global controllability analysis which make the closed-loop system x=

f{x,u(x))

locally asymptotically stable about some equilibrium state (x', u"), f(x*, u") = 0. Though a Lipschitz condition is only needed for many results stated here, we assume that / is C°° function of (x,u) and, for any fixed u £ R m , f(x,u) is a complete vector field on Rn. As we have already seen in Chapter 3, Part I, the equilibrium set /- x (0) = {(x,u) £ Rn x R m ;f(x,u) = 0} of the system S^ plays a crucial role in stabilization analysis and in stabilizer design. Definition 2.1 Let A be It is said that the system stabilizable at (x", u") ) if, u(x*) = u* and the point of the closed loop system

a class of admissible feedbacks, and(x',u*) £ / - 1 (0)£/ is locally stabilizable at (x*,ur) by A (or locally and only if, there exists a feedback u(x) £ A such that x" £ Rn is locally asymptotically stable equilibrium x=

f(x,u(x)).

We recommend the reader to compare this definition with Definition 3.2, Chap­ ter III, Part I. In this chapter we often will consider smooth stabilization at (i*,u*) or, in other words, C°° stabilization, i.e. A = C°° We recall the following classical result of stability theory. Theorem 2.1 ( A.Liapunov [29]) Consider a nonlinear system x = f(x),

/(0)=0,

xGRn

(2.1)

such that f(x) = Ax + o(x) where lirn^o o(x)/\x\ = 0 and A £ R

nxn

for

x —> 0,

. Then:

(i) if A is a Hurwitz matrix, i.e. det(sl - A) / 0

whenever

Re(s) > 0,

then the origin is a locally asymptotically stable solution for (2.1), Re(s) is the real part of a complex number s £ C;

215 Global controllability

and

stabilization

(ii) if there is at least one

eigenvalue

A 6 spA = {s 6 C; d e t ( s i - A) = 0} then the origin is not stable for

with

Re(s) > 0,

(2.1).

Using Teylor expansion, f(x, u) = f(x*, u*) + A{x-x*) of f(x,u)

at (x*,u*),

+ B(u -u*)

+ o(x-

x*,

u-u*),

where ,. lim

o(x — x*,u — u") —^ ——, '— = 0

ac-KC«,a-»u« \x - x*\ + \u — U*\

and A = | ( ^ « » ) ,

B = ^(.-lU-),

we can write the linearized system (2.2)

i = Az + Bv. We call the mode A 6 5pA not controllable (controllable) if rank(XI

— A, B) < n (rank(M

— A,B)

= n).

The following first-order sufficient and necessary conditions of local stabilizability are immediate consequence of Theorem 2.1. T h e o r e m 2.2 If the linearized system (2.2) is stabilizahle, rank(sl

— A, B) = n

whenever

i.e.

Res > 0,

then the original system E / is locally C°° stabilizahle at (x*,u*). hand, if there exists X £ spA such that Re\ > 0

and

rank(\I

— A,B)

then the system S / is not locally C°° stabilizahle at

On the other

< n, (x',u*).

If the linear system (2.2) is stabilizahle, then it is wellknown ( see, e.g. [39]) that there exists a linear feedback v = Fz, F G R m x n , which is stabilizer for (2.2). This linear feedback is also a local stabilizer of the nonlinear system S / . This can be proved by showing that a quadratic Liapunov function for the system x

=

Ax + Bv,

v

=

F(x — x*)

216 Global controllability analysis is also a Liapunov function for the closed loop system x = v =

f{x,v), F{x-x*).

For a proof, see e.g. [39]. It is easy to see that if T,f is C°° stabilizable at (i*,u*) then there exists a smooth feedback u = u(x) such that u(x') = u* and, for each z in a neighborhood of the point x" G R n , one can find a control «.(t) = u(e«'(z)), / = /(x,«(x)),

(2.3)

which steers the system S/ from z to i* as t —+ oo. Summarizing necessary conditions of local smooth stabilizability we obtain the following result. Theorem 2.3 ( R. W.Brockett [5]) If the system S/ is C°° stabilizable at (x',u*), then: (bi) all the modes of (2.2) with positive real parts are controllable; (bii) there exists some neighborhood Q of x' £ R such that for each y £ Q one can find a control uy(t) : [0, oo) -* Rm ■which steers the system S/ from y at t = 0 to x* at t = oo; (biii) the mapping f(x, u) : R£ x R^f -♦ Rn maps every neighborhood of (x", u*) onto a neighborhood of zero. Proof. Conditions (bi),(bii) are obviously true, (bi) follows from Theorem 2.2 and (bii) is the corresponding open-loop property (2.3) of a smooth stabilizer. To prove (biii) we use Brouwer theorem [31] and converse statements of Lia­ punov theorems [33]. Without loss of generality we assume that x" = 0, «* = 0 and u = u(x) is a C°° feedback law which locally stabilizes S/ at the origin. By the converse of Liapunov theorem [33], we have a C°° Liapunov function V(x) defined on a neighborhood Q of the origin such that V(x)>0

WxeQ,x^0;

V(x) = 0 for

x =0

and < gradV(x),f(x,u(x))><0

VxGQ,x^O.

217 Global controllability

and

stabilization

Since V(x) is continuous, there exist real numbers e > 0, h > 0 for which

{V(x) = h}cQ and < gradV(x),f(x, m

W GR ,f £ R

u(x) + v) + f > | K ( I ) = / , < 0 n

such that

|£| < e, \v\ < e.

Making use of Brouwer theorem [31], for each |u| < e, |£| < e, we obtain z £ {V(x) < h} such t h a t

f(z,u(z)

+ v) = -i,

Therefore the continuous mapping f(x,u) : R° x R|f —► R n maps every neigh­ borhood of t h e origin onto a neighborhood of zero. T h e proof is completed. Q.E.D. In mathematics it is commonly known t h a t only those necessary conditions are of interest to us t h a t are almost sufficient. Prom this point of view t h e conditions (bi) and (bii) are important b u t (biii) is not very interesting, because (biii) is almost always satisfied. In other words, (biii) is generic. T h a t means (biii) does give us a very little information about C°° stabilization. There was an a t t e m p t t o strengthen t h e condition (biii) ( see [10]). To formulate t h e main result of [10] we need some machinery of homology groups. T h e reader interested in homology and cohomology groups is referred to the books [12,48]. Others m y skip Theorem 2.4 and start to consider exam­ ples at t h e end of this section. For a topological space X and for a nonnegative integer q we denote by Hq{X) the q— t h singular homology group of M with coefficients in Z = {0, ± 1 , ± 2 , . . . } ( for an exact definition see, e.g. [12,48]). It is easy to see that /([R?xIC]\f-1(0))cRn\{0}, and therefore / induces a homomorphism / , : -ff„_i([R? X K\

\ f _ 1 (0)) -> H n _i(R" \ {0}).

Moreover, if v : R n —* R m is a C°° feedback law which locally stabilizes S / at (x*,u*), then, for a proof see, e.g. [12], ( / o (-, u ( - ) ) ) . ( F „ _ 1 ( R n \ x*)) = R W R " \ 0), where ( / o (.,»(■)))(*) = / ( * , » ( * ) )

Vi6Rn.

Since for any e > 0 there is S > 0 such that the diagram Fig.2.1, where

Bs(x') = {x e Rn; |x-x*|<*},

218 Global controllability analysis

Figure 2.1: The diagram is commutative. Qc(x*,u*) = {(*,«) g RJ x I C ; |x - x*| + |u - u»| < e}, is obviously commutative, we obtain the commutative diagram Fig.2.2. Thus we have proved the following Theorem 2.4 ( J.-M. Coron [10]) If the system Y,j is locally C°° stabilizable at a point (x*,u*) £ / _ 1 (0), then, for any e > 0,

/.(£r„_i(g«(*V) \ r x (o))) = ^«-i(Rn \ o) = z. This necessary condition is stronger than (biii), but nevertheless it is still generic, and therefore , fulfilled for almost every smooth nonlinear system S/. Thus the only reasonable necessary conditions of local C°° stabilization, that we have obtained until now are (bi) and (bii). Consider examples of systems which are controllable but not locally C°° sta­ bilizable. Example 2.1 ( R.W.Brockett [5] ). Let ii

=

m,

X2

=

"2,

X3

=

X2Ui — I i t / 2 .

It is easy to show that this system is controllable on R3. On the other hand, the necessary condition (biii) is violated, and therefore the system is not locally

219 Global controllability and stabilization

Figure 2.2: The diagram is commutative. C°° stabilizable at the origin. We recommend the reader to verify that after a perturbation which is small in the sense of C°°— topology of the space of smooth functions on a small compact neighborhood about the origin the system always satisfies the condition (biii). Because of this fact (biii) can not be considered as a "good" necessary condition of local C°° stabilization. Example 2.2 ( D.Aeyels [1]). Consider the system i\ X2

= =

=

X\ -f- 3?2; U. U.

The condition (bi) is void. It follows the system is not locally C°° stabilizable at the origin. On the other hand, applying Kawski procedure described in Chapter IV of Part I, this book, we can construct a Holder continuous feedback law which locally stabilizes the system at the origin. Example 2.3 (J.-M. Coron [10]) The system ii x2

= u2(xi — x2), = u2(x2 — u).

is controllable, since it is equivalent to the cart, see Chapter II of Part I. But the existence of C°° stabilizing feedback law at the origin is excluded in force of Theorem 2.4.

220 Global controllability analysis We leave for the reader to verify that under a perturbation, which is small in the sense of C°°— topology of the space of C°° functions on a compact neighborhood about the origin, any system satisfies the conditions of Theorem 2.4 and consequently this theorem still gives us a little information about local C°° stabilization.

2.2

Liapunov's direct method

Here we investigate local stabilization of nonlinear systems using Liapunov's direct method. Recall that a function V : R n —» R + is said to be a Liapunov function for a system * = «*),

« 0 ) = 0,

{eCfR-.R"),

(2.4)

1

if V(x) is a C function and there is a neighborhood Q of the origin such that V(x)>0

VxeQ,x/0,

V(0) = 0 and < gradV(x), £(x) > < 0 Vx 6 Q, x # 0.

(2.5)

The classical Liapunov's result ( see [29] ) states that if there exists a Liapunov function V(x), then the origin is locally asymptotically stable. Sometimes, it may be easier to find a positive definite function V(x) which satisfies
> < 0 Vx 6 Q

instead of (2.5). Such a function will be called a weak Liapunov function. Theorem 2.5 ( La Salle's Invariance Principle [3]) IfV(x) punov function for (2-4), the set Q£ = {x G Rn;

is a weak Lia­

V(x) < e}

is compact for some e > 0 and the set Q£ n {x g Rn;

< gradV(x), £(x) > = 0}

does not contain any nontrivial trajectory of (2.4), then the origin is asymp­ totically stable for (2-4)Using this principle we will prove several important for applications sufficient conditions of local stabilization.

221 Global controllability and stabilization

2.2.1

Sontag's formula of almost smooth stabilizer.

Henceforth in the rest of this chapter, for the sake of simplicity, we limit our consideration to the single input case of an affine nonlinear system having the form £ ( / , b): x = f(x) + b{x)u, u g R, where f(x), b(x) are complete C°° vector fields on Rn We assume also /(0) = 0, so that we can take u(0) = 0. Let us use from now on the notations a(x) =

L,V(x),

P(x) =

LbV(x),

where V{x) is a C°° function such that V(x) > 0 for x # 0 and V(0) = 0. Then Sontag's formula ( see also [40] ) f 0

if/9(*) = 0,

u{x) = | _ . ( . H y ( y W . ) £ .f ^

+ Q

(2.6)

gives us a stabilizing feedback law u(at) £ C°°(Rn \ 0) n C(R n )

(2.7)

when i / V(a;) < 0 for

r ^ 0 and

LfcV(a;) = 0.

(2.8)

Indeed, for the closed loop system we have < gradV(x), f{x) + u{x)b{x) >= - ^ / ( a ( i ) ) ' + (^(i)) 4 . The standard Liapunov's theorem [29] implies asymptotic stability of the origin for the system S ( / , 6) closed by the feedback law (2.6). Let us prove the regularity of u{x), i.e. the inclusion (2.7). It is easy to see that H(a, 0, z) = Pz2 -2az-p3 for (a,P) e S = R2 \ {(a,P); P = 0 and

=0

a > 0}, whenever

z = u{x) and u(x) is defined in (2.6). The condition (2.8) means (a{x),P(x))€S

Vx^O.

222 Globed controllability

analysis

On the other hand,

!■#(<»,&*)# o v( a>j 0)es. Thus, the implicit function theorem guarantees u(x) g C°°(R n \ {0}) D C ( R n ) where u(x) is given by (2.6). For a more complete proof the reader is refereed to the article [40]. In the paper [30] an analogous formula for a stabilizing feedback law using a bounded control is also provided.

2.2.2

Jurdjevic-Quinn approach.

Here we use a more Namely we say that Jurdjevic-Quinn type tive definite function

strong condition on the function V(x) instead of (2.8). the system S ( / , b) satisfies a Liapunov condition of the if there exist a neighborhood Q of the origin and a posi­ V(x) such that V(x)>0

VzgQ\{0}, V(0) = 0

and < gradV(x),f(x)

>< 0

for all

T h e function V(x) is called also a Liapunov function type for S ( / , b).

xEQof the

Jurdjevic-Quinn

The Jurdjevic-Quinn approach is based on the following theorem.

T h e o r e m 2.6 ( A.Bacciotti [2]) Consider the affine nonlinear control system S ( / , 6) defined by C°° vector fields f,b. Assume that / ( 0 ) = 0 and there exists a C°° Liapunov function of the Jurdjevic-Quinn type V(x) for E ( / , b) in a neighborhood Q about the origin. Then the C°° feedback u(x) = - < gradV(x),

b(x) >

(2.9)

locally stabilizes S ( / , b) at the origin if, and only if, there exists a positive real number e such that, no trajectory of the vector field f(x) is contained in the set Vc = {x G Q; V(x) = c, LbV(x) = 0} Vc g (0, e). Proof. Sufficiency of the proposition easily follows from Theorem 2.5. To prove necessity we notice that if, for some c g (0, e), t h e set Vc contains a trajectory of the vector field / ( x ) , then this is also the trajectory of the system £(/,&) closed by the feedback (2.9). Thus the proof is completed. Q.E.D.

223 Global controllability

and

stabilization

In their original paper [23] Jurdjevic and Quinn exclude a non trivial trajec­ tories of the vector field f(x) from Vc (c € (0, e)) by means of a condition expressed in terms of Lie brackets of vector fields / and b. Corollary lLetV(x) be a Liapunov function of the Jurdjevic-Quinn type, such that gradV{x) ^ 0 for each x ^ 0. Assume that rank{adkfb(x);

k = 0,1,...} = n

for each x ^ 0 in a neighborhood of the origin. by means of (2.9). Proof. Since the function LfV(x) x€Z

Then, the system is stabilizable

has a maximum at every = 0, LbV(x)

= {LfV{x)

= 0},

we obtain grad(LfV(x)) = 0 (2.10) for each x 6 Z. We have to prove t h a t Z does not contain any nontrivial trajectory of t h e vector field f(x). Indeed, if there exists x 6 2 , i / 0, such that etf(x) e Z for all t > 0, then (etf)'{LbV){x)

= 0

for all

t > 0,

and hence {jt)h^f)'(LbV)(x)\t=0

= 0

Vfc = 0,1,2,....

Thus, using (2.10) we obtain ( e 7 ) * i 6 V ( S ) = < gradV{x),etadfb{x)

>,

and hence Lad^V{x) But gradV(x)

= Q V* = 0 , l , 2 , . . . .

^ 0 and the contradiction obtained completes the proof.

Q.E.D.

2.2.3

Stabilization of homogeneous systems

The control system E ( / , b) is called homogeneous if f(x) and b(x) are homo­ geneous vector fields, i.e. there exist nonnegative real numbers k0 and ki such that / ( A s ) = A*»/(z), 6(Az) =

\hlb{x)

for all x € R n and A > 0. We use the notations

fco = deg(f),

224 Global controllability analysis fci = deg(b) and call fc0,&i degree of the vector field f(x) and degree of b(x), respectively. We consider here only homogeneous control systems such that deg(f) is an odd integer and deg(f) > deg(b) > 0. Definition 2.2 The system S ( / , 6) is said to satisfies the homogeneous Lia­ punov condition if there exists a homogeneous C°° positive definite function V with homogeneity degree deg(V) = fi > 2 such that V(Xx) = X"V(x) < gradV(x),f(x)

VA>0,

>< 0 for all x ft 0,

i£Rn, < gradV(x),b(x)

> = 0.

We choose the feedback u{x) = -K\x\'

< gradV(x), b(x) >,

(2.11)

where K G R, K > 0 and 5

= deg(f) - 2deg{b) - defl(V) + 1.

Thus u(x) is homogeneous of the degree <%(/) - deg(b). Theorem 2.7 ( A.BacciottifS]) IfT,(f,b) satisfies the homogeneous Liapunov condition, then, for sufficiently large K, the feedback law (2.11) stabilizes the system S ( / , b) at the origin. Proof, the derivative of V(x) with respect to the system closed by (2.11) is given by < gradV(x), f{x) + u{x)b{x) > = L,V(x) -

K\x\'[LbV(x)]2,

where u(x) is defined by (2.11). By definition deg(V) > 2, and therefore the derivative of V(x) with respect to the closed-loop system is a homogeneous function of degree deg(V) - 1 + deg(f) > 2. Consider the unit sphere S"'1 = { i £ Rn; \x\ = 1} which is decomposed into two disjointed parts, Sn'1 = S~US+, where S~ =

{xeS"-1;LfV{x)<0},

S+ =

{x€Sn-1;LfV{x)>0}.

225 Global controllability and stabilization The homogeneous Liapunov condition implies {x G S"~1\Li,V(x) = 0} C S~~. Assume that S+ ^ 0, otherwise the proof is trivially completed. Since both S+ and S" _ 1 are compact, there exist I = min(L 6 V(x)) 2

L = max LfV(x),

and clearly L > 0,1 > 0. We choose K > &. Thus if x G 5+, then < gradV(x)J(x)

+ u(x)b(x) >< L - Kl < 0,

where u(x) is given by (2.11), while if x G S~, then the derivative of V(x) with respect to the closed-loop system is also negative. Therefore, since this derivative of V(i) is homogeneous, it is negative for all i ^ 0 and the proof is completed. Q.E.D. This theorem can be generalized to the control systems having the form * = [/]*.(*) + - + [/]*.+/(*) + 1*,

(2-12)

where 6 G R" and [f]k0+i(x) {i = 0,.., /) denotes a vector field whose compo­ nents are homogeneous polynomials of the same degree k + i. Let P denote a positive definite symmetric real matrix and let Zo(P) = {xe Rn; < x,P[f]ko(x) Zi(P) = {xeRn;

>< 0} U {0},

< x , P [ / ] t o + , ( x ) > < 0 } , i = l,...,J.

We choose the feedback U (x)

; = - J r - ^ | x | 8 ' <x,Pb>,

(2.13)

i=0

where Si = k0 + i - l ,

i = 0,l,..,/.

Let [Pb]j_ denote the set of all vectors orthogonal to Pb 6 Rn, i.e., [Pb]± = {x G Rn;

<x,P6>=0}.

Theorem 2.8 ( A.Bacciotti [2]) Let k0 be odd. If there exists a positive defi­ nite symmetric real matrix P G R n x n such that

[Pb]±cZ0(P)nZ1(P)n...nZl(P), then system (2.12) is stabilizable by the feedback (2.IS).

226 Global controllability analysis Proof. Let S"" 1 denote the unit sphere. Since both S? = S"'1 \ Z,-(P) (t = 0,1,..., /) and S1"-1 are compact, there exist I = max{max < x,P[f]ko+i(x)

>; i = 0,1,

...,l},

1 = min{min < x,Pb >; i = 0,1,...,/} xesf

and L > 0,1 > 0. If we choose if > j , then the derivative of V(x) = < I , P I > with respect to the system (2.12) closed by (2.13) is negative for x ^ 0. Therefore the statement is proved. Q.E.D.

2.3

Decoupling normalizing transformations and local stabilization of nonlinear sys­ tems

In this section the existence of the normalizing transformations completely decoupling the stable dynamic from the center manifold dynamic is proved. Then a numerical procedure for the calculation of asymptotic series for the decoupling normalizing transformation is proposed. The developed method is applied to obtain some new sufficient conditions for local stabilization of nonlinear systems with noncontrollable linearizations.

2.3.1

Existence of decoupling normalizing transforma­ tions.

Consider the system x = Ax + <&(x,y), (2-14) y = By + V{x,y), where (x, y) £ Rm x Rn, A € R m x m and A =

Rn denotes n-dimensional Euclidean space, -AT,

the eigenvalues of B £ R n x n have negative real parts,

227 Global controllability and stabilization $ and \& , are at least C 2 functions which vanish together with their derivatives at the origin, i.e., *eC*(RmxRn,Rm),

$(0,0) = 0 ,

d$(0,0) = 0, (2.15)

k

m

* E C (R where k > 2, functions

n

n

x R , R ),

f (0,0) = 0,

d*(0,0) = 0,


^ r x R " - . R', which have continuous derivatives of order k. To investigate the dynamic of the system (2.14) in a neighborhood of the origin we apply the center manifold theory which mainly consists of the following three theorems.

Theorem 2.9 (J.Carr[8],A.Kelley [25]) Given the conditions (2.15), then there exists a center manifold Mc = { ( i , y ) e B { ( 0 ) x R " ; y = h(x)}, where Bs(0) = {x G Rm; | x |< <5}, X) XjZi

for

x, z e R', h 6 C

k_1

| x | 2 =< x,x > m

and

< x,z > =

n

(R , R ) and S is a sufficiently small real

i=l

positive number.

It is convenient to use the following notations:

J{x,y)= (Ax + $(x,y),By + V(x,y))T, e{f denotes the flow generated by the vector field / . etf(x,y) is the point drifted by the flow e*' in time t from the point (x, y). The zero solution is said to be stable, iff for every neighborhood W there exists a neighborhood V, such that et}V CW

V< > 0,

228 Global controllability

analysis

where t*'V = {e^{x,y); (x,y) 6 V}. T h e zero solution is asymptotically stable, iff it is stable and there exists a neighborhood S, such t h a t lim etf(x,y)

= 0

t—>+oo

for all (x, y) 6 S. T h e flow on the center manifold Mc is governed by t h e system (2.16)

z = Az + $(z,h{z)).

T h e next theorem tells us that (2.16) has all t h e necessary information needed to determine the asymptotic behavior of (2.14) in a neighborhood of t h e origin.

T h e o r e m 2.10 ( J.Carr[8]) (a) If the zero solution of (2.16) is stable (asymptotically stable) (unstable), then the zero solution of (2-14) i3 stable (asymptotically stable) (unsta­ ble). (b) If the zero solution of (2.16) is stable, then there exists a neighborhood V of the origin, such that for every (xo, j/o) £ V one can find ZQ, such that e ^ x o . t / o ) = (z(t,z0),h(z(t,z0))) where 7 > 0 is a constant, condition z(0, z0) = ZQ.

z(t,Zo)

+ 0(e-*),

is the solution

of (2.16) with

initial

The center manifold can be approximated to any degree of accuracy. For C 1 functions