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0: mm ,/,(<« TL de0ned by
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
with
i
w2
where W-2 is a subset of Q, fi, or <9fi, depending on the type of control.
This type of
approximation is called Tikhonov regularization [12]. For each j3 > 0, the coefficient to be identified, (/>, is viewed as a control and is adjusted to get the corresponding solution, «(), close to the observations z. For a fixed /? > 0, the optimal control, hp, that minimizes
Jf}(4>)^ will be explicitly characterized in terms of solutions of the optimality system, which consists of the state problem coupled with an adjoint problem. Taking a sequence of j3n that
converges to zero, the corresponding sequence of optimal controls, <j)pn, is shown to converge to a "solution", (/>*, of our identification problem. The interpretation of this "solution" is as follows: If the identification problem is known to have a solution, then our method will
find a solution to that identification problem. If the observations are imprecise and thus the identification problem may not have a solution, then our method finds a projection onto an appropriate range space. This interpretation holds for the case of one observation (one input yielding one output), but in the third example, we discuss its extension to multiple observations coming form multiple inputs. The majority of traditional approaches to inverse identification problems [1, 2, 3] couple Tikhonov's regularization with an optimization algorithm. Fuel and Yamamoto [10, 11, 13] have obtained uniqueness and stability results for reconstruction algorithms using exact controllability for various wave equation problems. Our approach has the advantage of an explicit characterization of the approximate coefficients; moreover, this characterization leads to a natural numerical algorithm in solving the resulting optimality system.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
1. FIRST EXAMPLE This example identifies the dispersive coefficient, h(x, t ) , from observations of the solution
of a wave equation on a set Q' C Q = £1 x (0,T). Consider the wave equation:
uu = AM + hu + f
Q
in
u = MO, M = MI
on
fi x {t = 0}
M = 0,
on
<9ftx(0,T),
(1-1)
where / e L 2 (Q), MO G /^(Q), MI e L 2 (O) and Q C M" has a C2 boundary. The identification problem is to find bounded h such that the corresponding solution u = u(h) of system (1.1)
is close to the observations z on Q'. We consider the control set,
U = {h e L°°(Q)| -M < h(x,t) < M}, and the approximate functional is:
Jp(h) = -2 ( j (u(h) - z)2 dxdt + (3 [ ti2 dx dt] .
\JQ'
JQ
J
We seek hp € U such that
See Liang [8] for some further results on this type of control problem. The joint work with Liang [9] gives more details on this particular example. The solution space is
ueL*(Q,T;Hb($l)),
uteL2(Q),
utt e L 2 (0,T; H^
From differentiating the maps
h —> u(h) and
h -+ J0(h), with respect to h and a priori estimates, we obtain
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Theorem 1.1. There exists a control hp £ U and corresponding state up = u(hp), that
minimizes the functional Jp(h]
over U.
Furthermore, there exists a weak solution p in
2
L (0,T;H^(n)) to the adjoint problem: ptt = Ap + hp + (up — Z)XQ'
in
Q
p =Pt=0,
on
ttx{t = T}
p= 0
on
<9fix(0,T),
(1.2)
where XQ' is the characteristic function of the set Q' , and, hp satisfies (1-3) This sequence hp of optimal controls approximates h, the desired coefficient as /3 —* 0.
If the measurements are inaccurate or affected by noise, then the observations z may either come from a solution of (1.1) that does not represent the actual scenario or from a function
that is not a solution of (1.1) for any h 6 U. Thus we do not assume that z is the range of the map (1.4) In this case we can prove:
Theorem 1.2. There exists:
(i) a sequence (3n —> 0 (ii) corresponding optimal controls, hpn, for the functionals Jpn(h),
(iii) h* e U and
(iv) u* = u(h*) such that hpn -^ h* and
u(h0n) —*• u"
L2(Q),
weakly in weakly in
L (0,T;//g(fi))
I (u - z)2dxdt = inf / (u(h) - zfdxdt.
I x-,<
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
hf JI
I ^,
Note the limit u* can be interpreted as a (not necessarily unique) projection of z onto the range of the map (1.4).
2. SECOND EXAMPLE In this example, we seek to identify the reflection coefficient a of part of the spatial boundary, dfl. For bounded domain D C R 2 with Cl boundary, define the spatial domain
n = { ( x , y , z ) \ ( x , y ) £D,w(x,y) < z < 0} where w : D —> (— oo,0) is a C2 function. Assume the region fi contains a certain medium (like water in a section of the ocean) and denote, as before, Q = fi x (0,T). We seek to identify the reflection coefficient from that set
< t7(x,y) < K}. Consider the solution, u = u(a), of the acoustic wave equation:
utt = Ait + /
in
Q
u = 0,
on
E x (0,T), sides of spatial domain
fs = 0
on
13 x {z = 0} x (0, T), top of spatial domain
|^j + au — 0
on
F x (0, T), bottom of spatial domain
u = UQ,
on
f7 x {0}.
ut = HI,
(2.1)
Where r={(x,y,w(x,y))\(x,y)eD}
E = { ( x , y , z ) \ ( x , y ) e dD,w(x,y) < z < 0}, (?1i
and 7— = Vw • 77 denotes the outward co-normal derivative. To approximate the identificaov tion problem, we consider the objective functional
(u-zfdxdydzdt
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
+ /3
(a(x,h))2
where G C fi is a set with positive measure. We seek to identify a from observations z on the set G x (0,T), resulting from a single source /. To define our solution space, let
V = {v <= Hl(ty\v<0
on E}
with norm
|Vi>|2 dxdydz
-1/2
The solution space for the state system (2.1) is defined by
e L\Q)
where V denotes the dual space of V. Assume:
G C fi
with positive Lebesgue measure
z e L 2 (G x (0, T)),
w e C 2 (D),
10(1, j/) < 0.
From [6, 7], we state the combined existence and characterization result for the approxi-
mate functional Jp(a).
Theorem 2.1. There exists a unique control o~0 in U and corresponding state up = u(crg),
that minimizes the functional Jp(cr) over U. Furthermore, there exists an adjoint solution
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
m L2(0,T;V) such that Pa = Ap + (up - h}xo
*n
Q
p=0
on
Ex(0,T)
|^ = 0
on
D x {z = 0} x (0, T)
f* + <W = 0
on
rx(0,T)
on
p = pt = 0,
(2.2)
n x {T},
and
r / i /-^
<rp(x,y) = mint - / upp(x,y,u(x,y),t) [ \P Vo
\+ i
dt \ ,K\. / J
(2.3)
VKe now /ef /? —> 0 and i/ie sequence of optimal controls <jp converge to the desired coefficient [7].
Theorem 2.2. Suppose the inverse problem has a solution, i.e., there exists a* G U such that u* = u(a*) satisfies u* = z a.e. on G x (0,T). Then there exists O~Q €. U such that on a
subsequence j3n —> 0, we have o~/3n —•*• O-Q
L2(T)
in
wpn = w((3n) -+ WQ in
L 2 (0,T; V)
and WQ = z
a. e. on
G x (0,T).
We also solved a more complicated problem along this line, namely we identified at the
same time the shape of part of the boundary, w(x,y), and the reflection coefficient, o~(x,y)
[5].
3. THIRD EXAMPLE We apply the optimal control techniques to reconstruct the dispersive coefficient in a wave
equation from a single input of Neumann data and possibly noisy observation (output) of
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Dirichlet data. For Q C K n and Q = fi x (0, T), consider the wave equation:
utt = Au + hu + f
in
Q
u = MQ,
on
(7 x {i = 0}
ut = MI,
(3-1)
Given Neumann data, 5 e Hl/2(d£l x (0,T)), we seek to identify the dispersive coefficient h, from observations z on <9fJ x (ij, i 2 ). This reconstruction is done from a single Neumann-
to-Dirichlet type measurement. The approximate control problem treats ft, as a control in
- M < H(x) < M} and seeks to minimize for J3 > 0,
Jp(h) = \( I (u(h) - z)2 ds dt + /? f h2 dx] . 2 \Jdnx(ti,t2) Jn J We make the following assumptions:
<9QeC 2 z e L 2 (9fi x (*i,t 2 ))
withO < ^ <£2 < T
The solution space for the state problem is
u£Hl(Q)
with
utteL^O.T;^1
We gave a complete characterization of the optimal control hp in [4] .
Theorem 3.1. There exists an optimal control hp and corresponding state, up = u(hp),
minimizing Jp(h) over U. Furthermore, there exists a solution p m Hl/2(Q) solving the
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
adjoint problem: pu = Ap + hpp
in
Q
p = Q,
on
n x {T}
pt = 0,
(3.2)
where X(ti,t2) ^ the characteristic function of the time interval (ti,^),
ana
hp = min ( max ( —- / upp(x, t) dt, - M ) , M ] . V V P Jo ) )
(3.3)
Note that the solution space for the adjoint problem is weaker than the state solution space due to the less regular Neumann data.
Under additional regularity assumptions of /, z, and g, we can prove the uniqueness of the
optimal control hp. Now as /3 —> 0, we do not assume the identification problem has a solution, which allows for noisy or inaccurate observations.
Theorem 3.2. There exists
(i) a sequence /3n —> 0 (ii) a sequence of corresponding optimal controls, hpn, for the functionals Jpn(h), (iii) h* e U and u* = u(h*), such that hn —^ h*
weakly in
u(hpn) -^ u"
weakly in
u((3n)tt-^ u*tt weak* m
Hl(Q)
L2 (0,T; Hl(Q)*)
and (u' - z) 2 ds dt = inf /
I Jdnx(ti,t2)
h&U
Jdflx(ti,t2)
We do not assume z is in the range of the maps NDh:
NDh : Hl NDh(g) =
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(u(h)-zfdsdt.
Thus the limit u* can be interpreted as a projection of z onto the range of these maps. Numerical illustrations related to Example 3 can be found in Ref. [4]. To consider multiple inputs g and corresponding observations z(g),
define the input set
G = {g 6 and the set of observations {Z 6
where K2 depends on K\ . We assume the map
9 € G -* z(g) is compact, i.e.
gn^g*
in
implies z(gn) —> 2(5*) in L?(d£l x (0,T)). We note that the solution u of the wave equation depends on h,g; i.e.,
Then we can prove:
Theorem 3.3. There exists u* e L 2 (0,T; H l ( f y )
with utt 6 L 2 (0,T; f/^Q)*), /i* e C/ and
g* e /f 1/2 (<9O x (0,T)), swc/i f/iat u* = u(h*,g*) and
f I J8nx(ti,t 2 )
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
f (u* — z(<7*)) 2 dsdt = minmin / 9eG
h€U
JdClx(ti,t2)
\u(h, g) — z(g)\'2 dsdt.
REFERENCES [1] Banks, H.T. and K.K. Kunish, Estimation Techniques for Distributed Parameter Systems, Birkhauser,
Boston, 1989. [2] Borggard, J. and J. Burns, PDE Sensitivity Equation Methods for Optimal Aerodynamic Design, ICASE Report 96-44, NASA Langley Research Center.
[3] Isakov, V., Inverse Problems for Partial Differential
Equations, Springer Veiiag, Berlin, 1998.
[4] Feng, X., S. Lenhart, V. Protopopescu, L. Rachele, and B. Sutton, Identification Problem for the Wave
Equation with Neumann Data Input and Dirichlet Data Observations, submitted to IMA Journal on Applied Math.
[5] Lenhart, S., V. Protopopescu, and J. Yong, Identification of Boundary Shape and Reflexivity in a Wave
Equation by Optimal Control Techniques, Diff. and Int. Eqs., 13(2000), 941-972.
[6] Lenhart, S., V. Protopopescu, and J. Yong, Identification of of a Reflection Boundary Coefficient in an Acoustic Wave Equation, ISAACS Conference Proceedings, Direct and Inverse Problems of Mathematical Physics, Kluwer Publishers, 2000, 251-266.
[7] Lenhart, S., V. Protopopescu, and J. Yong, Optimal Control of a Reflection Boundary Coefficient in an Acoustic Wave Equation, Applicable Analysis 69(1998), 179-194.
[8] Liang, M., Bilinear Optimal Control for a Wave Equation, Math. Models and Methods in Applied
Sciences, 9(1999), 45-68.
[9] Liang, M., S. Lenhart, and V. Protopopescu, Identification Problem for a Wave Equation via Optimal Control, Control of Distributed Parameter and Stochastic Systems, Kluwer Academic Press, Boston, 1999, 79-84. [10] Fuel, J. P., and M. Yamamoto, Applications of Exact Controllability to Some Inverse Hyperbolic Problems C. R. Acad. Sci. Paris, 320(1995), Series 1, 1171-1176.
[11] Fuel, J. P., and M. Yamamoto, On a Global Estimate in a Linear Inverse Hyperbolic Problems, Inverse
Problems 12(1996), 995-1002. [12] Tikhonov, A. N., and V.Y. Arsenin, Solutions of Ill-posed Problems, John Wiley, New York, 1977.
[13] Yamamoto, M., Stability, Reconstruction Formula, and Regularization for an Inverse Source Hyperbolic Problem by a Control Method, Inverse Problems 11(1995), 481-496.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Singular Perturbations and Approximations for Integrodifferential Equations .]. Liu, J. Sochacki, P. Dostert Department of Mathematics, James Madison University, Harrisonburg, VA 22807.
Abstract
Let e > 0 and consider
and
w ' ( t ) = Aw(t)+fb(t-s)Aw(s)ds Jo
+ f(t),
t > 0, tu(0) = wa,
in a Banach space X. Here the unbounded operator A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and b(-) is a continuous scalar function. We will look at the singular perturbations when £ —> 0, and approximate the above two integrodifferential equations with
two corresponding systems of ordinary differential equations for which the numerical solutions can be carried out easily. An application to partial differential equations with numerical solutions is given.
1
INTRODUCTION.
We study the integrodifferential equations
e2u"(t;e) + u'(t;e) = Au(t;e) + f^ b(t - s)Au(s;e)ds + f ( t ; e ) , u(0;e) = w 0 (e), u'(6;e) = u { ( e ) ,
t>Q,
(
,
, ' '
and
' w'(t) w(0)
= Aw(t) +^b(t-s)Aw(s)ds + f ( t ) , = WD,
t > 0,
^^
in a Banach space X, where the unbounded operator A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and &(•) is a continuous scalar function. We regard Eq.(1.2) as the limiting equation of Eq.(l.l) as e ~> 0. Now, Eq.(1.2) is of lower order in derivative of t, in this sense we say that we are dealing with the singular perturbation problems. In an early study in Liu [9], it was shown that under some convergence conditions on the initial data and /(i;e), one has u(-;e) —» w(-) as e —> 0. In this paper, we will apply the techniques we developed in [10] to approximate the two integrodiffernetial equations (1.1) and (1.2) with two corresponding systems of ordinary differential equations for which the numerical solutions can be derived easily. This way, we can provide a very useful procedure to numerically approximate the integrodifferential equation (1.1) arising from engineering with some simpler system of ordinary differential equations. Also, we will be able to see how the singular perturbations are carried out as e —> 0. An application with numerical solutions will be given to a partial differential equation.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
2
APPROXIMATION METHODS.
In this paper we make the following hypotheses ([6]) : (HI). The operator A generates a strongly continuous cosine family and a strongly continuous semigroup.
(H2).
&(•) e C2(R+,R), R+ = [0,oo).
(H3). f(--e) and f&Cl(R+,X),
£
> 0.
(H4). u 0 (e),w 0 £ D(A),u0(e) -» w0, £2ui(e) -» 0, as e -> 0. (H5). For any T > 0 , / ( • ; £ ) - > / ( • ) in L l ( { 0 , T } , X )
as e -> 0.
We say that u : R+ -> X is a solution of Eq.(l.l) if u £ C" 2 (fi + , X), w(t) 6 -D(A) (domain of A) for £ > 0 and Eq.(l.l) is satisfied on R+ . Solutions of Eq.(1.2) are defined in a similar way. Note that the existence and uniqueness of solutions of Eqs.(l.l) and (1.2) are obtained in [3, 5, 7, 14, 15], and we are only interested in singular perturbations and approximations in this paper, thus we assume that Eqs.(l.l) and (1.2) have unique solutions u(t;e) and w(t)
respectively for every e > 0, when the initial data satisfy certain conditions. From the singular perturbations results in [9], we have
Theorem 2.1. [9] Assume that hypotheses (HI) - (H5) are satisfied and let T > 0 be fixed. Then as e —> 0, one has u(t;s) -» w(t) in X uniformly for t € [0, T]. Next, we modify the techniques we developed in [10] so as to approximate Eq.(l.l) with a simpler system of ordinary differential equations. First, as in [10], we invert Au(-) so that Eq.(l.l) can be transformed into a form with continuous kernel, that is, the unbounded operator A will not appear in the integral. Theorem 2.2. Equation (1.1) is equivalent to
u(0;e)
) + f
+ f(t;?),
t > 0,
,„ n
'
where b(-) is a continuous scalar function determined by &(•).
Proof. Define
fl and 6*H = H. Jo Then we can find a C2 solution FofF + b + F*b = 0 (see, e.g., [2, 4, 8, 11, 12]) such that
R*H(t)= / R(t-s)H(s)ds
(6 + F ) * ( 6 + b)=6. Now. write Eq.(l.l) as
£2u"(£) + u'(e) = (6 + b)* Au(e) + f ( e ) . Then we have
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(6 + F ) * \e2u"(e) + u'( £ )] = Au(e) + (S + F) * f ( e ) .
(2.2)
Hence
£2u"(e) + u'(e) = Au(e] + (6 + F) * /(e) - F * [
Integration by parts yields /"' F*u'(t;s)= / Jo
ft
F * u"(t- e) = / F"(t - s)u(s; s)ds + F(0)u'(t; e) - F(/:) Wl (e) + F'(0)u(t; e) - F ' ( t ) u 0 ( e ) . Jo Therefore Eq.(l.l) can be replaced by
£2u"(t;e) + [l + £2F(Q)]u'(t;£)
= [A - F (0) - e2 F' (0)}u(t; e) [~F'(t - s) - £2F"(t - s)}u(s; £)ds
+ [(6 + F) * f ( t ; £) + F ( t ) u 0 ( e ) + e*F'(t)uQ(e) +e2F(t)Ul(£)}, (2.3) hence we complete the proof by dividing [1 + £2F(0)}.
D
Next, for Eq.(2.1), we approximate the continuous scalar function &(•) by a polynomial
Pn(-) of degree n, and then use v'n(t;e)
= Avn(t;e) + /„' Pn(t - s ) v n ( s ; e ) d s + f ( t ; e ) , = ) , v'n(Q;e) =
t>0,
,„ , '
(
to approximate Eq.(2.1). To do that, we need the following inequality. Lemma 2.1.[1] Let u ( t ) be a nonnegative continuous function for t > a, and suppose that
u(t)
k(t, s)u(s)ds + I ( •J Of
J Ct
h(t,s,r)u(r)dr}ds,
t > a,
*J Q
where C\ > 0 is a constant and k(t, s) and h(t, s, r) are nonnegative continuous functions for a < r < s < t. If the functions k(t, s) and h(t, s, r) are nondecreasing in t for fixed s, r, then
u(t) < d exp | / k(t,s)ds+ Ja
( Ja
h(t, s, r)dr\ds\,
t > a.
Ja
Now, we can prove the following Theorem 2.3. Let TO > 0 be fixed. For any 6 > 0, there is a polynomial Pn(-) of degree n such that for the solution vn(-\'e) of Eq.(2.4) and the solution u ( - ; e ) of Eq.(2.1) (or Eq.(l.l)),
one has, uniformly for 0 < e < K where K is any given constant, max \u(t-e)-vn(t;e)\\<6. te[o,T 0 ]
(2.5)
Proof. Note that A also generates a strongly continuous cosine family, which we denote by
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
C(-). Then we can use the results in [6] to get, for t > 0, u(t;e)
=
t
C-
+ { G(t - s; e) Jo
vn(t;e)
b(s - r ) u ( r ; ^)dr + /(s; e) \ ds,
(2.6)
=
\ fs .
r
i
+ / G(t — S',s)\l I L n\s — T)vn(r\£)dr -f j(s;e) J us, Jo Jo
(^-7)
where /?(•;?), G(-;e) are linear operators denned in [6] using the Bessel functions; and they have the following properties: For some independent constants M > 1 and a > 0,
(PI). \\G(t-m<Meat, t>0, e > 0 . (P2). ||e-'/2 , t>0, e > 0 .
Using (P2) and Lemma 2.1, we can verify that there is a constant C independent of n and i'such that ||w n (i;f)|| < C for 0 < e < K, n = 1,2,..., and t & [0,To]. (See the treatment of u(t;e) — v n ( t ; £ ) below for details.) Therefore, G(t-s;e)\L
(b(s-r)u(r;e)-Pn(s-r)vn(r;£))dr]ds\\ 7o ^ ' i
G(t - s; E) L\ T ({b(s - r)[«(r; e) - vn(r; e)] Jo + \b(s - r) - Pn(s - r)]vn(r\e)\dr\ds\\ o
t
rs
(2.8)
_
/ Me a( '~ s) |&(s - r) - Pn(s - r)|||n n (r;e)||rfr-ds o Vo ' /"s ~ / Me \b(s — ^)l!iw(r'£^) — v (T'Ef]\\drds
(2 9)
o ,/o
< MeoT°T02C max |6(/) - Pnn(l)\ WI i€(o;r0] ^ '
r r a
Jo Jo
Now, we can apply Lemma 2.1 with k(t, s) = 0 and h(t, s , r ) = Me a ( s " s ) |6(s~r)j to obtain \\u(t;e) - v n ( t ; e } \ \ <
\
—
( f1 r
~
i
max \b(l) - P n (/)|)Me aTo T n 2 C exp-^ / / Me a(t " s) |W.s - r)|drds }. 'elo.Tol 1 "/ I ./„0 ./n J ^0
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(2.11)
Therefore we can find Pn to approximate b to complete the proof.
D
Now, for this polynomial Pn(-) of degree n, the (n + l)th derivative is zero. So we are able to rewrite (2.4) as a system of ordinary differential equations. Theorem 2.4. Eq.(2.4) is equivalent to an ordinary differential equation
Y'(t) Y(0)
= GY(t) + H ( t ) , t > 0, = Y0,
(2.12)
on Xn+3, where
0 =i-4 P n (0)7 7^(0)7
G=
7 0 0 . . -Jj7 ^7 0 . . . 0 0 7 ... 0 0 0 ... . . . . . . . . . . . . 1) Pri"~ (0)7 0 0 0 ... P^'(0)7 0 0 0 ...
. 0 0 0 . . . 0 0
0 0' 0 0 0 0 0 , 0 0 7 0 7 0.
0 4r /(<•£)
o' 0
H(t) = 0 0
, Vo =
" uQ(e) ' ui(e] 0 0
0
0
with / the identity operator. And G generates a strongly continuous semigroup on Xn+z. That is, Eq.(2.12) is wellposed on Xn+3. Proof. In equation (2.4), define
(2.13) (2.14)
= v'n(t;e).
Then =
^'(«;e)
1 /•'
1
4
/
(2.15) (2.16)
P
n(t -
Now define
= \ Pn(t - s)yi(s;£)ds. Jo
(2.17)
Then
Pn(Q)yi(t;e)
^ \ P'n(t~
Jo
- e)ds.
(2.18)
Next, define = / P'n(t - S)yi(s;e)ds, Jo
(2.19)
to get t- e)
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(2.20)
Continuing in this way we obtain i~t
(2.21)
o
y'kfr?) = P(n
(2.22)
and finally, we have
yn+3(t;£) =
(2.23)
y'n+3(t;£) =
(2.24)
Therefore, Eq.(2.4) is equivalent to Eq.(2.12). Next, G generates a strongly continuous semigroup on Xn+3 by using the perturbation results from the semigroup theory. This completes the proof. D
Now we see that by applying Theorems 2.2 - 2.4, the solution of the integrodifTerential equation (1.1) can be approximated by y i ( - ; e ) = v n ( - ; e ] , the first component of the solution of the system of ordinary differential equations (2.12) when 0 < 1 < K, where K is any given constant. Next, the results concerning the singular perturbations in [9] implies that the solution u of Eq.(l.l) and w of Eq.(1.2) are almost the same when e ~ 0. Now, from [10], Eq.(1.2) can
be replaces by w'(t)
t > o, w(0) = w0,
= (A + b(0)I }w(t) - I F'(t-s)w(s)ds
(2.25)
where F is from (2.2) and J ( t ) is determined by f ( t ) . Also from [10], w(-) of (2.25) can be approximated by z i ( - ) , the first component of the solution of
Z'(t) Z(0)
= G,Z(i ) + F l ( t ) , t = Z0,
0 7 0 . . . 0 0
. 0 0 . . . 0 0
> o, (2.26)
on X n + 2 , iwith
" A + 6(0)7 Pn(0)I 73n'(0)7 Gi =
7 0 0 . . . 7 3 n (n " 1) (0)7 0 75n(")(0)7 0
. . ... ... . . . . . . ... ...
0 0• 0 0 0 0 0 0 . Fi(t) = 7 0 7 0.
r ~
-I
0 0
0 0 , 0 0
Z0 =
(2.27)
0 0
where Pn is a polynomial of degree n approximating — F'. Therefore, these results imply that for e small, z\ from Eq.(2.26) can be used to approximate u of Eq.(l.l). Summarizing the above results, we have the following methods to approximate the solution u of Eq.(l.l): If e is small, that is, if the singular perturbation is considered, then the first
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
component of the solution Z of Eq.(2.26) can be used. Otherwise, if e is not small, then the first component of the solution 1' of Eq.(2.12) can be used. In either case, we are able to approximate the integrodifferential equation (1.1) (which is hard to solve) by using a simpler system of ordinary differential equations.
3
AN APPLICATION.
Let us consider the following partial differential equation from engineering,
putt(t, x; p) + aut(t, x; p) = uxx(t, x; p) + JQ b(t — s)uxx(s, x; p)ds + f ( t . x; p), u(t,0;p) = u(t,l;p)=0, u(0,x;p) = u0(x;p), ut(0,x;p)=ul(x;p), t > 0, z e [ 0 , l ] ,
(3.1)
in L 2 [0,1], where u is the displacement of an object, p is the density per unit area, and a
is the coefficient of viscosity of the medium. Divide a and change variables if necessary, we
may assume that a = 1. Therefore Eq.(3.1) is given by Eq.(l.l) with c 2 = p and A = ^ with (domain) D(A) = Wg'2[0, l]fW 2 ' 2 [0,1]. Thus, the results in [9] implies that, with some convergence conditions on the initial data and /(i,:r;e), when the density p —> 0, solutions of (3.1) will converge to solutions of the "limiting" heat equation
awt(t,x) w(t,0)
= wxx(t,x)+f*b(t-s)wxx(s,x)ds + f ( t , x ) , = w(t,l) = Q. w ( Q , x ) = w o ( x ) , t > 0, x e [0,1].
Next, let's look at how to approximate equations (3.1) and (3.2) with systems of ordinary differential equations. First, from Theorem 2.2 and a corresponding result from [10], we
see that the integrals in Eq.(3.1) and Eq.(3.2) can be replaced by integrals with continuous kernels. Thus, without loss of generality, and also for computational reasons, we may consider the following equations with a continuous scalar kernel E ( - ) ,
•£) + ut(t,x;£) u(t,0;e) u(0,x;e)
= uxx(t,x-e) + J* E(t - s)u(s, x;e)ds + f ( t , x;e), = u(t,l;e) = 0, = u o ( x ; e ) , ut(0,x;s) = u\(x\ e ) , t > 0, x e [0,1],
(3.3)
and
wt(t,x) w(t,0)
= wxx(t, x) + fQ E(t — s ) w ( s , x ) d s + f ( t , x),
= w(t,l)=Q, w(0,x)=w0(x),
t > 0, x £ [0,1].
Now, applying Theorems 2.3 - 2.4 to Eq.(3.3), that is, let P n ( - ) be the approximation of £(•), we see that, for Y(e) = (yi(;-,e),y2(;-,£),...,yn+3(;-]e)) in Xn+\ Eq.(2.12) for Eq.(3.3) becomes
j-ty4(t, x- £}
= P^yi (t, x; e) + y5(t, x; e), (3.5)
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
with the corresponding initial and boundary conditions. Now the first component y i ( t , x ; £ ) of the solution can be used to approximate the solution u of Eq.(3.3). For Eq.(3.4), the kernel E ( - ) is the same as in Eq.(3.3), thus the Pn for Eq.(3.3) can be used as Pn for Eq.(3.4). Therefore, for Z = (zv(-, •), z 2 ( - , • ) , . . . , zn+2(-, •)) in -X"+ 2 , Eq.(2.26) for Eq.(3.4) becomes
(3.6) 1 1
— —
)
(0)2! (t, X) + Zn+2(t, X),
with the corresponding initial and boundary conditions. From [10], the first component Z i ( t , x ) of the solution can be used to approximate the solution w of Eq.(3.4). The discussion at the end of the previous section indicates that when £ is small, z\ from Eq.(3.6) can be used to approximate u of Eq.(3.3); otherwise, y\(e] from Eq.(3.5) can be used. To check these results out, we look at the following example. But note that the solution u of Eq.(3.3) is difficult to obtain, thus we will demonstrate that for e small, z\ from
Eq.(3.6) and y\(e) from Eq.(3.5) are close, which, according to Theorem 2.3, implies that z\ from Eq.(3.6) and u of Eq.(3.3) are close. Example. Let E(t) = sin t in Eq.(3.3) and in Eq.(3.4) and let f(t,x\e) = 0 in Eq.(3.3) and let f ( t , x) = 0 in Eq.(3.4). Next, let u0(x; e) = w0(x) = x(l - x), and let Ui(x; e) = 0. Then
(H4) and (H5) are satisfied. We use Pn(t) = Y^Lo1^2(~~^Yt2i+i)>. ( wnen n ig °dd) as the approximation of sin t. Now Eq.(3.5) becomes :\e]
j,y2(t,x;e)
— y2(t,x;£),
= 4? = yi(t,x;e) (3.7)
j-tyn+2(t,x-e)
=yn^
with the initial conditions
yi(Q,x;e)=x(l-x),
yj(Q,x;e) = 0, j = 2,3, ...,n + 3, x e [0,1],
(3.8)
and the boundary conditions y j ( t , Q ; e ) = y^t.l'.e) = 0, j = l,...,n + 3, t > 0. Similarly.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Eq.(3.6) becomes ,x) ,x)
=z3(t,x), = zi(t,x) + z 4 (i,x)
(3.9)
with the initial conditions ~ ^U, (() X n.\ — X[i ~.("1 — X ~.\ Zi J — J,
r, ^ (0 — U, n Z ^ U , ™\ Xj —
— Z 9, 3 9, jn — O,...,n / t "-4-T ^
T tC X [ uffl, I 1 J1,
f^ 1 fTl ^O.-LUJ
and the boundary conditions Zj(t, 0) = Zj(t, 1) = 0, j = I , . . . , n + 2, t > 0. In the following, we solve Eq.(3.7) and Eq.(3.9) numerically. We use Forward Euler in t and Centered Differences in x, and apply the modified Picard method presented in [13]. Therefore Eq.(3.7) becomes yi,k(tj+i,Xi;c)
= yi,k(tj,Xi-s) + /^J+1 y 2 , k - i ( s , X i ; c ) d s ,
y2,k(tj + ltxi'>~)
=
y2,k(tj, Xi',s) + ;
(Ax)
-y2,k-i(s,Xi,E) + y3,k~i(s,x,;e) ds,
Vn+3,k-ls,
and Eq.(3.9) becomes
Z2,k(tj+l, Xi)
= Z2je
= Zn+1,k(t.j, Xi) +
j+l
2:n+2;Jt-i(s, Xi)ds, 1
^Xi) +ff>+ (-l)(»-WZltk_1
(s,Xi)ds.
In the figures below we present solutions to Eq.(3.7) and Eq.(3.9) using the above Picard iterations with n = 5, xt = iAx, i — 1, ...,20, Ax = 0.05, tj+i - tj — At = 0.00125, and let
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Fig. 1 e = 0
0.251
Fig. 2 e = 0.015625
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Fig. 3 e = 0.0625
References [1] D. Bainov and P. Simeonov, Integral inequalities and applications, Kluwer Academic Publishers, Boston, 1992. [2] W. Desch, R. Grimmer, Propagation of singularities for integrodifferential
equations, J.
Diff. Eq., 65(1986), 411-426. [3] W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integrodifferentid equations, J. Math. Anal. & Appl., 104(1984), 219-234. [4] W. Desch, R. Grimmer and W. Schappacher, Propagation of singularities by solutions of second order integrodifferential equations, Volterra Integrodifferential Equations in Banach Spaces and Applications, G. Da Prato and M. lannelli (eds.), Pitman Research Notes in Mathematics, Series 190, 101- 110. [5] W. Desch and W. Schappacher, A semigroup approach to integrodifferential in Banach space, J. Integ. Eq., 10(1985), 99-110. [6] H. Fattorini, Second order linear differential land, 1985, 165-237.
equations
equations in Banach spaces, North - Hol-
[7] R. Grimmer and J. Liu, Integrodifferential equations with nondensely defined operators, Differential Equations with Applications in Biology, Physics, and Engineering, J. Goldstein, F. Kapple, and W. Schappacher (eds.), Marcel Dekker, Inc., New York, 1991, 185-199.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
[8] G. Gripenberg, S-O. Louden and O. Staffans, Volterra integral and functional equations, Cambridge University Press, Cambridge, 1990.
[9] J. Liu, Singular perturbations of integrodifferential equations in Banach space. Proceedings of the American Mathematical Society, 122(1994), 791-799. [10] J. Liu, E. Parker, J. Sochacki, A. Knutsen, Approximation methods for integrodifferential equations, Proceedings of Dynamic Systems and Applications, Vol. Ill, to appear.
[11] R. MacCamy, An integro - differential Math., 35(1977), 1-19.
equation with application in heat flow, Q. Appl.
[12] R. MacCamy, A model for one - dimensional nonlinear viscoelasticity, Q. Appl. Math., 35(1977), 21-33. [13] E. Parker and J. Sochacki, Implementing the Picard iteration, Neural, Parallel & Scientific Computations, 4(1996), 97-112.
[14] K. Tsuruta, Bounded linear operators satisfying second order integrodifferential tions in Banach space, J. Integ. Eq., 6(1984), 231-268.
equa-
[15] C. Travis and G. Webb, An abstract second order semi - linear Volterra integrodifferential equation, SIAM J. Math. Anal., 10(1979), 412-424.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Remarks on Impulse Control Problems for the Stochastic Navier-Stokes Equations J.L. MENALDI Wayne State University
Department of Mathematics Detroit, Michigan 48202, USA ([email protected])
S.S. SRITHARAN US Navy
SPAWAR SSD - Code D73H San Diego, CA 92152-5001, USA (e-mail [email protected])
Abstract
In this paper we will review certain recent developments in impulse control
problems for the stochastic Navier-Stokes equation. The dynamic programming equations for the optimal impulse control problem arises as a quasi-variational inequality in infinite dimensions which is resolved in a weak sense using the
semigroup approach.
1
Introduction
During the past decade several fundamental advances have been made in optimal control of fluid mechanics by a number of researchers [6]. In this paper we study impulse control theory for turbulence. In optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations. Variational technique to treat impulse control problems has been adapted to Gauss-
Sobolev spaces (e.g., Chow and Menaldi [2]) with partial results. However, because of the technical difficulties associated with the domain of the generator we prefer to follow
the semigroup approach. The dynamic programming approach is used to discuss a simple optimal stopping
time problem for the Navier-Stokes equation. We are forced to use sufficiently weak conditions on the data because our final objective is the optimal impulse control problems. In order to facilitate the use of the semigroup technique we first consider the 2-D
Navier-Stokes equation with random (Gaussian) forcing field. Several approaches have been proposed in the literature (see Sritharan [7] for a complete reference list). We then proceed to treat the infinite dimensional quasi-variational inequality to deal with the optimal impulse control problem in a weak sense. Complete proofs of the results stated in this paper can be found elsewhere (cf. [4] and [5]), here only the main ideas are given.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
2
Stochastic 2-D Navier-Stokes Equation
Let O C R 2 be a bounded domain with smooth boundary and u the velocity field. The Navier-Stokes problem can be written in the abstract form as follows
in L 2 (0,T;V),
dtu + Au + B(u) = f
(2.1)
with the initial condition
u(0) = u0
in H,
(2.2)
where UQ belong to H and the force field f is in L 2 (0, T; H). Let us begin by defining some standard function spaces,
V = {v € ej(0, R 2 ); V • v = 0 a.e. in O},
(2.3)
with the norm
v =
o
and H is the closure of V in the L 2 -norm \ 1/2 |v 2dx j = v . o
(2.5)
We will also define the following linear operators Pm : L 2 (C>,R 2 ) —> H 2
is the Helmhotz-Hodge orthogonal projection and
2
A : H (O,R ) n V —> H,
Au = -//P^Au, v > 0, is the Stokes operator, ^ ' '
where v is the coefficient of kinematics viscosity. the nonlinear operator
^ r D s C H x V —> H ,
The inertia term is represented by
5(u,v) = Pm(u • Vv),
(2.7)
with the notation B(u) = B(u, u). The domain of B requires that (u • Vv) belongs to the Lebesgue space L2(O, R 2 ). Let us consider the Navier-Stokes equation subject to a random (Gaussian) term i.e., the forcing field f has a mean value still denoted by f and a noise denoted by G. We can write (to simplify notation we use time-invariant forces) f (t ) = f (x, t) and the noise
process G(t) = G(x,i) as a series dGk - Y ^ k S k ( x } t ) d w k ( t ) , where g = (gi,g2, • • • )
and w = (u>i, 102, • • • ) are regarded as ^ 2 -valued functions. The stochastic noise process represented by g ( t ) d w ( t ] = ^kgk(x,t)dwk(t,u>) is normal distributed in H with a
trace-class co- variance operator denoted by g 2 = g 2 (t) and given by '(g2(t)u,v)= Tr(g 2 (*))=
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
We interpret the stochastic Navier-Stokes equation as an Ito stochastic equation in variational form
d ( u ( t ) , v) + (Au(t) + B ( u ( t ) ) , v) dt = (f , v) dt + £(g fc , v) d w k ( t ) ,
(2.9)
k
in (0, T), with the initial condition
(u(0),v) = (u0)v),
(2.10)
for any v in the space V. A finite-dimensional (Galerkin) approximation of the stochastic Navier-Stokes equation can be defined as follows. Let {ei,C2, • • • } be a complete orthonormal system (i.e., a basis) in the Hilbert space H belonging to the space V (and L 4 ). Denote by H^ the n-dimensional subspace of HI and V of all linear combinations of the first n elements {ei, 62, . . . , en}. Consider the following stochastic ODE in R."
(u"(f), v) + (Aun(t) + B(un(t)), v) dt = k in (0,T), with the initial condition (u(0),v) = (u 0 ,v),
(2.12)
for any v in the space Mn. The coefficients involved are locally Lipschitz and we need some a priori estimate to show global existence of a solution un(t) as an adapted process in the space C°(0,T,Hn).
Proposition 2.1 (energy estimate). Under the above mathematical setting let
Let un(t) be an adapted process in C°(Q,T,Mn) which solves the stochastic ODE (2.11). Then we have the energy equality
'd\un(t)\'2 + ^V |Vu"(Of <& - [2 (f(0,u"W) + Tr(g 2 (t))] dt + k
which yields the following estimate for any e > 0 rT
<|u(0)!
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
E{\Vun(t)\*}e~etdt < 2
,T
,
(2-15)
for any 0 < t < T. Moreover, if we suppose
then we also have fT
E{ sup \un(t}\pe-st+pv I 0
\Vun(t)\2\un(t)\p~2e~etdt}
<
JO
/n
-| y \
/O
for some constant Cef,T depending only on e > 0, 1 < p < oo and T > 0. Proposition 2.2 (uniqueness). Let u be a solution of the stochastic Navier-Stokes equation (SPDE) with the regularity ueL2(n;C'0(0,T;e)nL2(0,r;V)),
u e L4(O x (0,T))
(2.18)
and let the data f, g and UQ satisfy the condition
f 6 L 2 (0,T; V), 2
J/v zn L ( f t ; C ° ( Q , T , H )
g G L 2 (0, T;^(H)),
u 0 € M.
(2.19)
2
n L (0,T,V)) zs another solution then
u ( t ) - v(t)\2 exp f - v ^ / ||u(5)|| 4 4 ds] < |u(0) - v(0)| 2 , L Jo i- (°) J
(2.20)
u)zi/z probability 1 /or any 0 < i < T. Proof. Indeed if u and v are two solutions then w = v — u solves the deterministic equation
in L 2 (0,T;V),
dtw + Aw = B(u) - B(v) and settingO r(t) = ^f L ||u(s)| 4 \ / ^ JO It ^ ' ' L
4
(O)
ds we have
and integrating in i, we conclude.
D
Each solution u in the space L 2 (H; L°°(0, T; M) n I 2 (0,T;V)) of the stochastic Navier-Stokes equation actually belongs to L 2 (ft; C°(0, T; M) n L 4 (O x (O,! 1 ))) in 2D, O C E2. Thus in 2-D, the uniqueness holds in the space L 2 (ft; L 2 (0, T; V)). If a given adapted process u in L 2 (fi; L°°(0, T; H) n L 2 (0, T1; V)) satisfies for any function v in V and some f in L 2 (0,T;V) and g in Z, 2 (0,T; £ 2 (H)), then we can find a version of u (still denoted by u) in L 2 (ft; C°(0, T; H)) satisfying the energy equality
d|u(0| 2 = [2{f(0, u(0) + Tr(g 2 (i)] dt + 2(g(<), u ( f ) ) dw(t) e.g. Gyorigy and Krylov [3].
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(2.22)
Proposition 2.3 (2-D existence). Let f, g and UQ be such that
f eLp(0,T;V'),
gGLp(0,T;£2(H)),
u 0 € H,
(2.23)
/or some p > 4. TVien i/ie?'e is are adapted process u(t,x,u>) with the regularity
u e L P (Q; C°(0, T; H)) n L 2 (0; £ 2 (0, T; V))
(2.24)
which solves the stochastic Navier-Stokes equation and the following a priori bound holds r E{ sup u(t)\p+ I o
Vu(t)\2 u(t)\p~2dt} <
/or some constant Cp — C(T,v,p) depending only on the numbers T > 0, v > 0
P>
2.
a
The proof of this result can be found in our previous work [4, 7]. Our proof is based on the L 4 -monotonicity of the nonlinear Navier-Stokes operator (and it generalizes to other cases, including multiplicative noise). If we denote by Br the (closed) L 4 -ball in
r},
(2.26)
then the nonlinear operator u i—> Au + B(u) is monotone in the convex ball Br i.e., QO r 4
(Aw, w) + (B(u) - 5(v),w) + —— |w|2 > - H w l l 2 ,
(2.27)
Vu 6 V, v e Br and w = u - v.
3
Markov-Feller Process
In what follows for the sake of simplicity we assume that the processes f ( x , i , w ) and
g(x,i,w) are independent o f t , i.e., f € V
and
g 6 4j(H)
(3.1)
and we denote by u(f; UQ) the semiflow, i.e., the solution of Navier-Stokes equation.
Also usually we substitute UQ with v.
Proposition 3.1 (continuity). Under the previous conditions the stochastic semiflow u(t; v) is locally uniformly continuous in v, locally uniformly for t in [0,oo). Moreover,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
for any p > 0 and a > 0 there is a positive constant A sufficiently following estimate
large such that the
E{e-at(X + \u(t; v)| 2 ) p / 2 } < (A + v 2 ) p / 2 , Vi > 0, v G H holds, also for any stopping time t = T. Furthermore, iff
(3.2) and g belong
to H and ^(V) respectively, then the semiflow is also locally uniformly continuous in i, locally uniformly for v in V. D
The Navier-Stokes semigroup ($(*), t > 0) defined by $(t)h(v) = E{h(u(t;v))}, is indeed a Markov-Feller semigroup on the space Cb(W) (of continuous and bounded real function on El endowed with the sup-norm). Since the base space H is not locally compact, the Navier-Stokes semigroup is not strongly continuous. In our approach, it is convenient to work with unbounded functions. Let CP(W) be
the space of real uniformly continuous functions on any ball and with a growth bounded by the norm to the p > 0 power, in another words, the space of real functions h on H such that v i—> h(v)(l + |v| 2 )~ p/ ' 2 is bounded and locally uniformly continuous, with the weighted sup-norm
where A is a positive constant sufficiently large to so that
a > a 0 (p),
p > 0.
(3.4)
It is clear that Cb(M) = C0(W) and Cq(W) C C P (H) for any 0 < q < p. Then for any a > 0, (linear) Navier-Stokes semigroup ($„(<), t > 0) with an a-
exponential factor is defined as follows
$„(*) : CP(H) —— CP(W),
M<)A(v) - E{e-ath[u(t; v)]}.
(3.5)
Proposition 3.2 (semigroup). Under the above assumptions the Navier-Stokes semigroup ($„(£), t > 0) is a weakly continuous Markov-Feller semigroup in the space
<7P(H).
a
Since the Navier-Stokes semigroup is not strongly continuous, we cannot consider
the strong infinitesimal generator as acting on a dense domain in CP(M). However, this Markov-Feller semigroup ($ 0 (t), t > 0) may be considered as acting on real Borel functions with p-polynomial growth, which is Banach space with the sup-weighted norm and denoted by BP(W). It is convenient to define the family of semi-norms on Bp(W) s>0
(s-v))\e^°s},
VveH,
(3.6)
where A is sufficiently large. Now, if a sequence {hn} of equi-bounded functions in BP(M) satisfies po(hn — /i,v) —> 0 for any v in H, we say that hn —> h boundedly pointwise convergence relative to the above family of semi-norms. It is clear that Po($a(t)h — h, v) —> 0 as t —> 0, for any function h in C'P(H) and any v in EL
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Definition 3.3. Let C'P(H) be the subspace of functions h in BP
such that the
mapping given by t H-> h[u(t\ v)] is almost surely continuous on [0, +00) for any v in and satisfies
\\mpo(
Vve
where p o ( - , •) is given above.
D
This is the space of function (uniformly) continuous over the flow u ( - , v ) , relative
to the family of semi-norms and it is independent of a. Hence, we may consider the Navier-Stokes semigroup on the Banach space C'p(H), endowed with the sup-weighted norm. The weak infinitesimal generator — Aa with domain T>p(Aa) (as a subspace of C'p(H)) is defined by boundedly pointwise limit [h — $a(t)h]/t —> Aah as t —> 0, relative
to the family of semi-norms. Notice that p0($a(t)Ji, v) < p0(h,v) for any t > 0, Ji in CP(W) and v in M.
Proposition 3.4 (density). // the above assumptions hold, then CP(K) C CP(S), the Navier-Stokes semigroup leaves invariant the space CP(S) and for any function h in Cp(Irl), there is _a equi-bounded sequence {hn} of functions in the domain T>p(Aa)
satisfying po(hn — /z, v) —» 0 for any v in EL
D
^From above results it is clear that given a > 0, p > 0, A sufficiently large and a function h in CP(W) there is another function u in T>p(Aa) such that Aau = Ji, where
$a(t)hdt.
(3.8)
The right-hand side is called the weak resolvent operator and denoted by either Tia = A~l or *R,a — (Aa + Q/)™ 1 . Moreover, if a0 = «o(A) then for any p > 0 we have a 0 (A) —> 0 as A —> oo, and for any stopping time r, |Vu(f; v)| 2 (A
ur;
2p 2
/
Vv G H, and then for any a > a0 we obtain
||<MO/»|| < £-(«-«»)' P||,
po(
(3.10)
for any t > 0, and
\\Ka~h\\ < for an
v in
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
a — QO and where the
a — a0 || and the semi-norms p o ( - , v).
(3.11)
4
Impulse Control Problem
Let us now consider the problem of sequentially controlling the evolution of the stochastic process u(t; v) by changing the initial condition v. For this purpose we consider a controlled Markov chain qi(i) in H with transition operator Q(k) and a control parameter k which belongs to a compact metric space K. For a sequence ((",-, i = 1 , 2 , . . . ) of independent identically distributed H-valued random variables we have
),
' '"' ' Vv e H,
(4.1)
for any initial value q(l), any bounded and measurable real-valued function h on HI and any k in K. For the sake of simplicity, this Markov chain (i.e., each random variable (",) is assumed to be independent of the Wiener process w — (wi,Wz,. • •) used to model the disturbances in dynamic equation. A sequence {T;, fc,-; i = 1 , 2 , . . . } of stopping times T,- and decisions fc; such that r,
approaches infinity is called an impulse control. At time t =• T; the system has an impulse described by the (controlled) Markov chain qjt(z') with k — A:;. Between two consecutive times T, < t < r,-+i, the evolution follows the Navier-Stokes equation: f u ( i ) = uft.Ti-uM),
<
[and
if
Ti < « Ti+1,
~
(4.2)
u(r;) =q(u(r t —),C; A:;),
where u(t, s; v) is the solution of Navier-Stokes eqaution with initial value v at time s. Since T,- —> oo, we can construct the process u(i) by iteration, for any impulse control {r,-, fcj-; z = 1 , 2 , . . . } and initial condition v in H. It is clear that r,- is an stopping time with respect to the Wiener process enlarged by the cr-algebras generated by the random variables Ci, ( 2 , . . . , Ci-i- Also, the decision random variables A;,- are measurable with respect to the cr-algebra generated by T,-. To each impulse we associate a strictly positive cost known as cost-per-impulse and given by the functional L(v,k). The total cost for an impulse control {T,-,^,-; i = 1 , 2 , . . . } and initial condition v is given by oo
/
F(u(t))e-aidt + Y] L(u(Tt-), k,)e-aT'} ~^
(4.3)
and the optimal cost t / ( v ) = inf J(v,{r,-,fc,-}),
(4.4)
{r,,kt}
where the infimum is taken over all impulse controls, and u(t) is the evolution constructed as above, with initial condition v.
Let us follow a hybrid control setting as in Bensoussan and Menaldi [1]. The dynamic programming principle yields to the following problem. Find U in CP(M) such that
U < MU,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
AaU < F,
and
AQU = F
in
[U < MU],
(4.5)
where Aa is interpreted in the martingale (semigroup or weak) sense and M is the following nonlinear operator on C(1HI) given by
Mh(v) = m f { L ( v , k ) + Q ( k ) h ( v ) } ,
Vv e H,
(4.6)
where Q(k)h(v) = E{h(q(~v,(^i | k ) ) } is the transition operator. This problem is called a quasi-variational inequality (QVI). To solve the QVI we define by induction the sequence of variational inequalities
(VI) (Un+l £ C'p(M) n+1
(AaU
such that
=F
m
n+1
[U
Un+l < MUn ,
AaUn+l < F
and
n
< MU ],
where U° — U° solves the equation AaU° — F. This VI can be formulated as a maximum sub-solution problem
Un+l e CP(H)
such that
Un+l < MUn,
AaUn+l < F,
(4.8)
for any n > 0. In view of the Theorem for the VI in [5], we need only to assume that M operates on the space Cp(lrl) to define the above sequence U" of functions. This means that first, we impose the condition
(\\L(-,k)\\
VkeK,
V v € H, and next
t4'1^
I lim s u p { p 0 ( $ a ( t ) Q ( k ) h - Q ( k ) h , v ) } = 0,
y k e K, v € H, V h e C"p(H), for any m > 0, some positive constant Cm and where the norm || • || and the semi-norms po(-, v). One of the main differences between impulse and continuous type control is the positive cost-per-impulse, i.e., the requirement
L(v, k) > t0 > 0,
Vv e H , k e A',
(4.11)
which forbids the accumulation of impulses. We also need
F e C'p(H),
F(v) > o, Vv e e,
(4.12)
to set up the above sequence. An important role is played by the function U° — U° which solves AaU° = F, and by the function UQ = UQ. which is defined as the solution of the following variational inequality
' U0 € Cp(lHI)
[aU0 = F
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such that
in
UQ<miL(-,k), k
[U0<mtL(;k)],
AaU0 < F
and (4.13)
or as the maximum sub-solution of the problem
t/o 6 C P (H)
such that
U0 < inf L ( - , k),
AaU0 < F.
(4.14)
Consider the quasi-variational inequality (QVI) f t / £ Cp(Irl)
such that
[AaU = F
in
U < MU,
Ajj < F
and
[U<MU],
or the maximum sub-solution of the problem
U e Cp(H)
such that
[/ < Mf/,
A,t/ < F.
(4.16)
Theorem 4.1 (QVI). Let the above assumptions hold. Then the VI defines a (pointwise) decreasing sequence of functions Un(v) which converges to the optimal cost U(v), for any v in H. Moreover, if the condition
there exists
r £ (0,1}
such that
r U°(v) < U0(v),
Vv £ H
(4.17)
is satisfied then we have the estimate
0 < (jn - Un+l < (1 - r)n L>°,
Vn = 0 , l , . . . ,
(4.18)
i/ie automaton impulse control {f,-, fc;}, generated by the the continuation region [U < MU} and defined by f0 = 0,
Ti = M{t> f,-.! f/lu^ju^!))] - Mf/[u(«;u(r,-_ 1 ))] }, fcE A
zs optimal, i.e., U(v) = J(v, {f,-, &,}), arac? ^/te optimal cost U is the unique solution of the QVI or the maximum sub-solution of the problem. D If we impose
L(v, k) > I0(l + v| 2 ) p/2 > 0,
Vv e H, jfc e K,
(4.20)
then assumption (4.17) holds for any 0 < r < 1 such that r \\F\\ < I0(a — a 0 )-
References [1] A. Bensoussan and J.L. Menaldi, Hybrid Control and Dynamic Programming, Dynamics of Continuous Discrete and Impulsive Systems, 3 (1997), 395-442. [2] P.L. Chow and J.L. Menaldi. Variational Inequalities for the Control of Stochastic Partial Differential Equations, in Proceedings of the Stochastic Partial Differential Equations and Application II, Trento, Italy, 1988. Lecture Notes in Math., 1390
(1989), 42-51.
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[3] I. Gyongy and N. V. Krylov, On stochastic equations with respect to semimartin-
gales Ito formula in Bariach spaces, Stochastics, 6 (1982), 153-173. [4] J.L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes Equation, Appl. Math. Optim., submitted. [5] J.L. Menaldi and S.S. Sritharan, Impulse Control of Stochastic Navier-Stokes Equations, Nonlinear Analysis, Methods, Theory and Application, submitted.
[6] S.S. Sritharan, (Editor) Optimal Control of Viscous Flow, SIAM, Philadelphia, 1998. [7] S.S. Sritharan, Deterministic and Stochastic Control of Navier-Stokes Equation with
Linear, Monotone and Hyper Viscosities, Appl. Math. Optim., 41 (2000), 255-308.
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Recent Progress on the Lavrentiev Phenomenon with Applications Victor J. Mizel
Carnegie-Mellon University, Philadelphia, Pennsylvania
The Lavrentiev phenomenon is associated with the sensitivity of the infimum of a variational problem to the smoothness of the class of admissible functions. Since the determination by Tonelli that the class of absolutely continuous functions is an appropriate class of admissible functions with which to obtain existence of minimizers to variational problems on a real interval by direct methods, it was shown that many of the classical problems yield to this approach. Moreover, in many of the classical problems the minimizers were actually Lipschitz or better, so that the problems were insensitive to the particular subclass of absolutely continuous admissible functions chosen for applying the direct method for existence. Therefore it was quite surprising when in 1926 M. Lavrentiev published, in response to a challenge issued by Tonelli during a lecture in Moscow, an example [L] of a functional of the form
J
,6
[y] = JIa f(x,y(x),y'(x))dx
with
y- K b\
subject to certain constraints and smoothness conditions, in which the infimum of J over the class of all absolutely continuous functions subject to certain boundary conditions at a and b is strictly smaller than its infimum over the class of all Cl functions
meeting the same boundary conditions. Thereafter in 1934 B. Mania published an example [Ma] involving a much simpler (polynomial) integrand. The presentation of an example involving a functional ,7 possessing a strictly elliptic integrand occurred only in 1985 [B&M] during the course of an investigation stimulated by the possible relevance of such questions to the theory of (multidimensional) hyperelastic materials. The present article reports on recent progress in the study of this phenomenon. We adopt the following notations for clarity. For each p £ [1, oo] Wl'p[a,b] = {y : [a,b] -> R\y G AC[a,b], ij G L p (a, &)}, so that for example with [a, b] = [0,1] and y(x) = x0, ft G (0,1), y G W^O,!] <^> p G [1,1/(1 - /?)). For J as above we put i(p] = mf{J[y]\y G Wl'p[a,b\ + EC's], p G [l,oo], so that Pi < "Pi =*• i(p\) < i(pi)- Then if / is such that i(p\) < i-(pi) for some pi < p? we
have the Lavrentiev Phenomenon A. (Cf [Bu&M],[Bu&B] for a relaxation view.) I. The first topic we study is a description of the possible boundary conditions which can lead to A, as well as the issue of whether this can occur when the integrand f in J is strictly elliptic and coercive. To facilitate this discussion we introduce for a,b,A,BE.M., p £ [1, oo] the following notation: (i) Both ends pinned [Lagrange problem] A-2(p) =
{yeW^[a,b]\y(a)=A,
y(b) = B}
(ii) One end pinned Aj(p) = {y G Wlf[a,b}\y(a) = A}
(iii) No ends pinned Ao(p) = Wl*[a, b], and we denote the respective infima by i - i ( p ) , i \ ( p } , and ?'o(p). The matter will be clarified by consideration of the following: 'Research partially supported by the US National Science Foundation.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
A. We take as our integrand f(x,y,z) = (if — x3)'2(l + z"20) so that J[y] = /oV-^ 3 ) 2 (l + (2/)20KT. and we take y e A:(p) = {y 6 W^[Q, l ] \ y ( 0 ) = 0}. Clearly, J[y] > 0, while for yo(x) = x3/5 J[y0] = 0. Hence by the computation on the preceding page, i\(p] = 0, if p E [I, 5/2). We now show that i](5/2) > 0 whence A occurs. The argument consists in treating two cases (cf[Ml],[DHM]):
Case 1: For some x* & ( Q , l ] , y ( x ) < [i(z)3/2]1/5, x <= (0,x*),y(x*) = [f (z*)3/2]1/s. Then by use of Holder's and Jensen's inequalities and the chain rule J[y] > k > 0, for k = (2/3) 20 (l/2) 6 ; Case 2: y(x) < [^(x) 3 / 2 ] 1 / 5 for all x £ (0, 1}. In this case a direct computation yields, with k as above, J[y] > k > 0. Although the integrand / is convex in z ["elliptic"] it is neither strictly elliptic nor coercive in z, but use of a device from [B&M] leads to the construction of a perturbed integrand f*(x,y,z) = f(x,y,z) + £22 which satisfies both requirements and retains the Lavrentiev phenomenon A i*(p) < z*(5/2), for all p e [1, 5/2).
B. With / as above we take J~(y] = j'^(rj5 - z 3 ) 2 (l + (?/)20)cte, y e Ao(p). By considering separately Case 1 y(0) > 0, Case 2 y(0) < 0, one deduces that A z~(p) = 0, if p 6 [1, 5/2), % (5/2) > k > 0 (with k as in A). Once again one can construct a modified strictly elliptic coercive integrand /* with the Lavrentiev phenomenon A i*0(p) < £[(5/2), p e [I, 5/2) (cf. [DHM]). II. The second topic concerns the role of A for autonomous integrands /, for integrands f which have no y-dependence, and for integrands which are jointly convex in (y,z). For first order problems it has been shown under various regularity assumptions that A cannot occur in any of these cases (cf. [C&V],[A&C], [Da], [S&M]).
Therefore we begin by considering an autonomous / which involves second order as well as lower order derivatives. We take as our integrand f(x,y,z,w) = f(y,z,w) — [(§2/)2 ~ (z + 4)2(|(-2 ~ 6))3]2-u;16, so that with obvious notation for the derivative of
the functions y'&AC[0, 1} J[y] = /^[(fy) 2 - (y' + 4) 2 (|(y' - 6)) 3 ] 2 (y") 16 dx, ,42>2(p) - {y 6 W^[0,l}\y(0) = 0,^(0) = 6,y(l) = 7,^(1) = f }. Clearly J[y] > 0, while J[yo] = 0, for yo(x) = 6x + x5/3. A direct computation shows that 2/o € W^fO, 1], p € [1,3). However it can be shown by the use of Holder's and Jensen's inequalities and the chain rule (cf. [M1],[H]) that i^(3) > 0. Thus we do have A ii(p) = 0,p E [1,3), ^2(3) > 0 in this autonomous second order context. Here, too, one can construct a modified strictly elliptic coercive autonomous integrand /* which retains the Lavrentiev phenomenon A (cf [H]). On the other hand, one can raise the question of whether in higher dimensions and first order variational problems any one of the three restrictions on the integrand mentioned above retains its role of excluding A. To the contrary it can be shown that there is an integrand / = f(F) for which all three restrictions mentioned above hold, such that the corresponding variational functional exhibits A with an elementary set of boundary conditions. This example is very closely related to that discussed in IV below (cf. [FHM2] for a detailed description). III. The next topic concerns the possible structure of infimum functions i(-) : [l,oo] —> [0,oo) exhibiting the Lavrentiev phenomenon. Although the simplest examples discussed above possessed right continuous jump discontinuities, it is not
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
evident that this is the only possibility. We consider first the following
f ( x , y, z) = (y2 - I6x)2{cz6 + [exp(-^;-1/2)2/2 - 16 exp(-x-1/2).T/(log.'r)2]2 exp(13z8)} foixe [0,l/e] Thus we have
J[y]= /01/e(2/2 for y e A-i(p) = {y E W*
this satisfies A z 2 (j>) = 0,p G [1,2), i 2 (2) 6 (0,1), i 2 (p) = l,p e (2,oo), so that z 2 is neither right nor left continuous at the jump point p = 1. Again it is possible to produce a modified strictly elliptic coercive integrand /* for which the lack of one sided continuity of i% at p = 2 is retained (cf.[S] for details and related results). Next we consider a two-dimensional example. Given po,pi G (1,°°) with Po < pi and a function a € W1>1[1, oo) satisfying
a(p) = Q,p £ [l,p()],a'(p) > Q,P 6 \po,P\},a(p) = a(pi) > 0,p e bi:°°); consider the integrand
f(x,y,u, z) = c(y)o'(y)|u|(m-3")/(»-1)(|«|''/(''-1) - x)2 z m withm e [3Pl,oo),c(y) =(m-3)(m-2)/2 [((m-l)/m)(j//(y-l))] m , a; 6 [0,1], y e [p0,Pi] and take
J[M] = /E
c(y)a'(y)\u (m-^/(y-^(\u\yl(y-^
- x) 2 |w, x \mdxdy, with
EPO,P1 = [0,lfx[po,Pi] and « 6 A(p) = {u 6 ^^(Epo^JKO, •) = 0,«(!,-) = 1}. The infimum for this problem satisfies A i(p) — a(p),p £ [1, oo), so that the infimum function can be absolutely continuous. The argument makes use of a 1988 result (cf. [H&M]) involving Noether's theorem for invariant one- dimensional variational problems (cf. [F] for details and related results). An effort to present a one-dimensional variational problem possessing properties of both types encountered above is currently under way ([M2]). IV. Our final discussion has to do with the topic which originally motivated the work by Ball and Mizel on the Lavrentiev phenomenon (cf. [B&M]), namely the relevance of this issue to the behavior of multidimensional nonlinearly elastic materials. Here we consider a homogeneous material in two dimensions with stored energy integrand W = W€ : Lin+(K 2 ) -> [0, oo) of the form
W(F)
= (||F||2 - 2detF)4 + c[(det^)-1 + (1 + ||F||2)"/2], for q > 2,e > 0, where
f
Lin (M 2 ) C Lin(R 2 ) denotes the set of linear operators on 2-dimensional space with positive determinant. It is easy to verify that W has the following properties: (a) W is smooth, (b) W is objective and isotropic, i.e. W(QF) = W(F) = W(FQ) for each orthogonal operator Q E Lin + (M 2 ), (c) W is polyconvex, i.e. there is a convex mapping g : Lin(K 2 ) x (0, oo) —•> R
such that W(F) = g(F, detF), for F e Lin+(M 2 ), (d) W(F) -^ +00 as detF -» 0 + , (e) W ( F ) > c , | | ^ | | ' - c 2 , c i > 0 , (f) W(F) = V(«i,w 2 ) = h-'>2 8 + £ [ l / ( u i V 2 ) + (l + (wi) 2 + (u2) 2 )« / 2 ], where the v's are the singular values of /'' [principal stretches of a deformation]. Thus the function A defined by A(6) = i/>(\/S, \/fi) is convex throughout (0, oo).
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
The variational problem to be considered involves the total stored energy
E[u] = jn W(Vu)dx for fi = {x e K 2 |zi 2 + z 2 2 < 1, xz > 0}, with u e .A(p) = {u e W^^R^Vu G Lin+ a.e. w^O) e [0,1] x {0},Xi 6 [0,1]; w(a:i,0) 6
{0} x [0,1], .T! e [-l,0];M(x) = (cos(9/2),sm(6/2)),x
= (cos0,sm0),0 e [0,7r]},
p > 2, so that w maps the unit upper half disk fi into the unit quarter disk fi'. We have the following result Theorem There is an £o( 0 such that if e < £o(g) and 2 < pi < 4 < p2 then
A i(pi) = inf{£[w]|w e -A(pi)} < i(pi) = \nl{E[u]\u e A(pi}} Notice that by the Sobolev imbedding theorem the displacements in both A(p\) and A(pi) are continuous on fi, so that no cracks are created. Indeed, the convexity of A in (f) ensures that the stored energy of continuous deformations from fi into fi' cannot be lowered even by enlarging class A to include discontinuous deformations of bounded variation with the same average density (cf. [M3]). That is, this material has the property that opening "cracks" in a deformation cannot result in a lowering of the energy. The proof depends critically on the following properties of the integrand W0 [i.e.,
W£ with e = 0]: (i) Wo > 0 is convex on Lin(M 2 ) (ii) solutions of the Euler-Lagrange system associated with WQ are known explicitly via complex analysis. In fact, u*(x) = [(a?]) 2 + (x 2 ) 2 ] 1 / 4 (cos(6»/2), sin(0/2)), where 0 is the polar angle to x, yields Wo(Vw*) = 0 and u* e A(p), for p e (2,4), while
u**(x) = ((x^ + (a; 2 ) 2 ] u / 28 (cos(
and _£/o[M**] > 0, with u* and u** both satisfying the Euler-Lagrange system for E = EQ (cf. [FHM1] for further details and related results).
Bibliography [A&C] G. Alberti and F. Serra Cassano, Nonoccurence of gap for one-dimensional autonomous functionals, Istituto di Matematiche Applicated, Universita di Pisa (1994)
[A&M] G. Alberti and P. Majer, Gap phenomenon for some autonomous functionals,
J. Conv. Anal. 1 (1994), 31-45. [B&M] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose mini-
mizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985) 325-388. [Bu&M] G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110 (1992) 434-460. [Bu&B] G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon in the calculus of variations, in Recent Developments in Well-Posed Variational Problems, R. Lucchetti and J. Revalski eds.. Kluwer 1995. [Ce] L. Cesari, Optimization-theory and Applications, Springer-Verlag, New York
1983.
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[C&V] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985)
73-98. [Da] A. M. Davie, Singular minimizers in the calculus of variations in one dimension, Arch. Rational Mech. Anal. 101 (1988) 161-177. [DHM] K. Dani, W.J. Hrusa and V. J. Mizel, On the Lavrentiev phenomenon for totally unconstrained variational problems in one dimension, Nonlinear Diff. Equations
and Appls. (in press). [F] M. Foss, Examples of the Lavrentiev phenomenon with continuous Sobolev exponent dependence, Center for Nonlin. Anal. Dept. Math. Sciences CMU Res. Report OO-CNA-013 (10/2000). [FHM1] M. Foss, W.J. Hrusa and V.J. Mizel, Occurrence of the Lavrentiev phenomenon in two dimensional nonlinear elasticity (in preparation).
[FHM2] __________________, On types of integrands exhibiting the Lavrentiev phenomenon in dimensions greater than one, (in preparation). [H] W.J. Hrusa, Lavrentiev's phenomenon for second-order autonomous variational problems in one dimension (preprint). [H&M] A.C. Heinricher and V.J. Mizel, The Lavrentiev phenomenon for invariant variational problems, Arch. Rational Mech. Anal. 102 (1988) 57-93.
[L] M. Lavrentiev, Sur quelques problemes du calcul des variations, Ann. Mat. Pura Appl. 41 (1926) 107-124. [Ma] M. Mania, Sopra un esempio di Lavrentieff, Boll. Un. Mat. Ital 13 (1934)
147-153 . [Ml] V.J. Mizel, New developments concerning the Lavrentiev phenomenon, Technion 1998, in Calculus of Variations and Diff. Equations Chapman and Hall/CRC
Research Notes in Math. #410,2000, A. loffe, S. Reich, I. Shafrir, eds. [M2] __________________, The Lavrentiev phenomenon in one dimension with general monotone exponent dependence (in preparation). [M3] __________________, On the ubiquity of fracture in nonlinear elasticity,
J. of Elasticity 52 (1999) 257-266. [S] A. Siegel, Two examples of Lavrentiev's phenomenon, Master's Thesis, Dept.
of Math. Sciences, Carnegie Mellon U. 1999. [S&M] M. Sychev and V.J. Mizel, A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational
problems, Trans. Amer. Math. Soc. 350 (1998) 119-133.
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Abstract Eigenvalue Problem for Monotone Operators and Applications to Differential Operators Silviu Sburlan Department of Mathematics, Ovidius University, Bd. Mamaia 124, 8700-Constantza, Romania E-mail: <[email protected]> Abstract
In this work we extend the multiple orthogonal sequence method, developed in [3], to the energetic space of an abstract linear monotone operator. This method leads to an abstract eingenvalue problem that it produces orthonormal bases in some nested Hilbert spaces, that they are suitable to develop abstract Fourier or projection methods. Some examples to diceerential operators are also given. One also considers the abstract semilinear eingenvalue problem and some results,
known for compact operators, are extended for monotone type operators with application to diceerential operators. Let X be a real Hilbert space with inner product (-, •) and the induced norm 11-11Consider a linear operator B : D(B) C X —»• X, with D(B) in0nite dimensional, which is symmetric, i.e., (Bu,v) = (u,Bv), Vu,v e£>(.B)
(1)
and strongly monotone, that is, there exists c > 0 such that
(Bu,u)>c\\u\\2, Vu£D(B)
(2)
We induce on D(B) the energetic inner product
(u,v)E := (Bu,v), Vw,v <E D(B) and the energetic norm
\U\\E := (u,u)lj>\ Denote by E the completion in X of the linear subspace D(B) with respect to the energetic norm an call it the energetic space of the operator B. It contains all u G X that are the limit points of Cauchy sequences {un} C D(B) with respect to the energetic norm || • ||.E. Extending by continuity the energetic inner product, i.e.,
(u,v)E :=lim(un,vn)E , Vu,v e E,
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the energetic space E becomes a real Hilbert space containing D(B] as a dense subset and the embedding E <—+ X is continuous, namely
The duality map J : E —+ E* , de0ned through
< Ju,v >:= (U,V)E, Vt/,v e E, is a linear homeomorphism with
\\Ju\\E. = \\U\\E, VueE, (see D. Pascali and S. Sburlan [7, p. 112]) and it is an extension of B, i.e.,
Ju = Bu, Vw £ D(B). The Friederichs extension A : D(A) C X —*• X of the operator B is de0ned through Au := Ju, Vw € D(A), (3) where D(A) := {u 6 E\ Ju £ X}. Observe that u 6 D(A) if and only if there exists an / G X such that
<Ju,v>=(f,v)E,
Vv&E
and D(B) C E C X C E* , (see E. Zeidler [11, p. 280]). Remark that the Friederichs extension is in fact the maximal monotone
extension of B in X since D(A) is dense in X and A is closed, self-adjoint, bijective and strongly monotone, i.e.,
(Au,u)>c\\u\\2, VuGD(A), (see A. Haraux [2, p. 48]). Also, the inverse operator A~l : X —>• X is linear continuous self-adjoint and compact, whenever the embedding E <^ X is compact. Therefore applying the Fredholm theory we can state following variant of the multiple orthogonal sequence theorem (G. Morosanu and S. Sburlan [3]):
Theorem 1: If the embedding E t—>• X is compact, then there exists the sequences {en} C E and {A n } C (0,+oo) that are eingensolutions of A, i.e.,
(Aen,v) = Xn(en,v),\/v£X, and such that:
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n&N
(4)
(i) (ii) (in) (i)
{en} {\/A n e n} {A n e n } {A n }
is an orthonormal basis in is an orthonormal basis in is an orthonormal basis in is increasingly divergent to
E; X; E*; +00.
Proof, (i) Suppose that the embedding E <^-> X is compact and let {un} 6 E be a bounded sequence, ||w n ||,E < c. Then, passing eventually to a subsequence, we can suppose that un —> M in X,and u £ E, because {un} is strongly and also weakly convergent in E. Then by the compacity of the embedding p-1^ - A-lu\\E = sup{< A~l(un - u},v>;\\v\\E < 1} = = sup{(u n - M,u)|H| B
A~len - knen, Vrz e N, form a Hilbertean basis in E (see e.g. S. Sburlan & co [9, p. 186]). Hence
Aen - \nen, Vn <E N where the characteristic values, \n = •£-, are such that \n < Xn+i —*• +00, as stated in (iv), and therefore (4) holds. iii) To show that {V%Ten} is a Hilbertean basis in X it suffices to prove that it is complete in X because &mn = (e m ,e n )fi = A n ( e m , e n )
(5)
as it results from (4). For this let h G X be such that
(h, v^en) = 0, Vn e N
(6)
by the Lax-Milgram theorem, with
a(u, v) := (u, V}E, Vu, v G E, it exists only one u £ E such that
(u,v)E = (h,v), \fveE Therefore, from (6) and (7), it results
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(7)
and, since {en} is complete in E, we have the implication u = 0 =>• h — 0, that is, the completeness in X of the system {\fX^en}. (iii) To show that {A n e n } is a Hilbertean basis in E* we must prove that it is an orthogonal system complete in E*. Note that E* is a Hilbert space with the norm
and the inner product de0ned by
For orthogonality we use the equation (4), namely
\^nen)\Sn) '—< ^n6n,en >= (A n e,j,e n J = (Aen,en) = = (e n ,e n ) B = \\en\\E = 1
and
/ii „ \ „ \ —(j-i(\\^ncf n)^ \ *>r-i(\ P \\^— \AnKn i A e )£,, — {J \Am^m))E — m
m
= (en,em)E — 0, Mm ^ n. For completeness let v* 6 E* be such that (v*, A n e n ) £ , = 0 for all n £ N. It then results (v*,h)E' = 0 , Mh &X because {\/\^en} is a complete system in X. Since X is dense in E* , for all € > 0 there exists an h£ 6 X such that £
> |K - Ml, = IKIII- - 2(«*,MB. + IIMI- = IHII* + ll^lll..
Hence, we have the implication |K|||. < £ , V e > 0 = ^ * = 0
as required.
•
Direct consequence: Denote by En := 5p{ei,e2, ....,e n } C £", Xn = = 5p{v / Aie 1 ,v / A262, ••-, V^ n e n } C X and £"* := 5p{Aifii, A 2 e 2 , ...,Xnen} C C E* , the 0nite dimensional subspaces generated by the 0nite sequence {e\, 62, ..., en}. Then En,Xn and E*n are projectionally complete in E,X and E* , respectively, that is Trnu —> u in each space, where n
irnu := ' a k < f > k , Vn > 1, with {
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
These coordinate systems can be used either for abstract Galerkin projection method or for abstract Fourier series method (see S. Sburlan and G. Moroanu
[10]). Remark: Since D(Bk) C D(Bk~l) C ... C D(B) and Bk is also symmetric we can apply the above method to produce orthogonal basis in the energetic
spaces associated to Bk, k G N, and the corresponding dual spaces. De0ne the spaces Ek '•= D(Ak/2), A; G N, where A is the Friederichs extension of B. It is easily seen that Ek are Hilbert spaces with the inner products: Vu,ue£fc. Of course, El = E and (u, u)i = (u, v)E, Vw, v G E (see E. Zeidler [11, p. 296]). As above we can easily show that {A^~~ + " en} is an orthonormal basis in Ek and
(8)
Since {\n} is an increasing sequence of positive numbers it results, by (8), that the embedding Ek+i ^ Ek is continuous. Also, identifying X by its dual space we get the inclusions
...- C Ek+l CEkC...CE1=ECXCE*=E^C
... C E\ C E*k+1 C ...
As the embedding E <^-» X is compact, it then follows that the embeddings Ek+i °—> Ek, X <—>• E* and E^ c—>• E^+1 are also compact. The duality mapping Jk : Ek —>• £"| is de0ned by < JkU, v >— (u, v ) k , Vu,v£Ek and, obviously, J\ = J. It is easily seen that P IN. M ,, Vn v u, kK, t
(£>} \y)
and then {A^ + 1 - ) / e n } is an orthonormal basis in E^.
Example 1. Let fi be a bounded domain in W1 with its boundary d£l enough smooth to apply the Green's formula. Take X := L 2 (f2), that is a Hilbert space with the inner product
( u , v ) := I u ( x ) v ( x ) d x , \/u,v G n and let B := —A be a de0ned on
D(B) :- {u e C 2 (fi) n Cl(ti)\u = 0 on F C dtt, //(F) > 0}. Then B is symmetric and strongly monotone having
E:= {v eH1(fy\v
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
= Qon F}
as energetic space. Note that E is a Hilbert space with the inner product
r
(u, V)E '•— I Vw(x) • Vv(x)dx, Vw, v 6 E J
n and the embedding E ^-> X is compact when N > 2 (see J. Neas [6,p.117]). Therefore we can apply the Theorem 1 and write (4) in the form Ve n (ar) • Vv(x)dx - \n I en(x)v(x)dx,
n
Vv £ E
(9')
a
Then by Green's formula we deduce that en are the weak solutions of the following eingenvalue problem
-Aen(x) = Xnen(x),x 6 fi, en(x) = 0 ,x£T,
(10)
Remark the limit case F = (ft when E = Hl(£l) is a Hilbert space with another inner product
f (u, V)E '•— I [u(x)v(x) + Vw(ar) • V v ( x ) ] d x , Vw, v e E that it modi0es (9') as follows / Ve n (ar) - Vv(x)dx = (A n - 1) / e n (x)w(a;)d a;
n
(11)
J
n
and, thus, e n are the weak solutions of the problem
-Ae n (ar) = (A n _i)e n (z), xett, ^(x) = 0 , x G 9J2.
(Uj
Application: The deformation of a body B, that occupies a bounded region f2 C ]$>N (N = 2 or 3), is characterized by the displacement vector u ifi^IR^ and
the corresponding strain tensor s = e(u). In the case of the small deformations, e(u) reduces to the symmetric part of the displacement gradient, i.e., l<M-<^}.
(13)
The constitutive relation that characterizes the elasticity is a generally nonlinear dependence of the stress tensor
on the strain, namely
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
a := {o-y |ffy = ff,-,-, 1 < i, j < N}
(14)
a = a(s) = A£ + o(s2),
(15)
where A :— {a^j/ 6 K|a,jfc/ = a,jik\ = akuj , I < i,j,k,l < N} are the elastic coefficients.
In the linear case, adopting the Einstein's summing convention
(i.e., the repeated index means summing over that index), we have the following relations (16)
known as the Hook's law. The coefficients A must depend continuously on the point in 17 or they are constants (the hyperelastic case) and they satisfy the ellipticity condition
aijk,£ij£kl > c\e 2 , Ve e RN*N,e = £T ,
(17)
where |-| means the Euclidean norm.
Let X := [L2(Q)]N be the Hilbert space of square integrable vectorial functions endowed with the inner product
[ (u,v) = / UiVi J
n and de0ne on
D(B) :={ve[c 2 ( the operator B of the linear elasticity
B\ = — divcr(£(v)), where the components of
• vdx =7,
o / akiij£ij(v)£ki(u)dx = -
aijki£ki(u)£ij(v)dx
=
n
diva(£(v))udx =: (u,Bv), Vu,v € D(B).
n Moreover, we have (5u,u) = - / aijki£ij(u)£ki(u)dx
n
> - I £ij(u)£ij(u)dx
= c I |Vu| 2 cfe.
n
n
The energetic space E is the completion on D(B) with respect to the norm I f
I
*• and it contains all vector functions from [//^(fi)]^ that vanishes on T. Moreover, the norm (18) is equivalent with the following one
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(18)
and thus, by Sobolev-Kondrashov theorem, the embedding E <^-> X is compact. Hence the duality mapping J : E —* E* is de0ned by
1 r < Ju,v>=- I aijkieij(vi)£ki(v)dx fi
(19)
and, by the Theorem 1: I f - I aijk,£ij(en)£k,(v)dx
f
= \n I (en)iVidx.
«/
n
n
By Green's formula we deduce that en are the weak solutions of the eingenvalue problem: —divu(£(en)) = A n e n in fi,
en = 0 on T,
(20)
•
Consider now the semilinear eingenvalue problem in X Lit + yuTV(w) = /,
(21)
where L G L(X) is compact and positive, TV is a nonlinear perturbation of Leray-
Schauder type (i.e., TV :— I — T with T a compact operator), and / G X a given element. Since L is, in particular, hemicontinuous and monotone it is maximal monotone and, thus, / + AL is invertible for each A > 0 and ||(7 + AL)" 1 1| < 1 (see e.g. D. Pascal! and S. Sburlan [7, p.106]). Consequently we can write (21) equivalently as
(I-M(X))u
=g
(22)
where M(A) := (/ + AL)~ : (7 — TV) is a compact operator, A = — G ffi+ and
g = (7 + \L)~lf. This it allows to introduce the coincidence degree for the pair (L, TV), simply, as follows: If D C X is an open bounded set such that
N(u) + XLu ^ f V u G 3D, A G M +
(23)
then the coincidence degree of the pair (L, TV) in D relatively to / is de0ned by
dx ((L, TV), D, /) := dLS (I - M(A), D, g } ,
(24)
where d^s denotes the Leray-Schauder degree. Of course, this degree has all the properties of Leray-Schauder degree, because of the above mentioned equivalence, and we can apply all classical results to (21). We point out here some in the linear case (see Mortici [4]): Suppose that TV G L(X). We say that A G M+ is a characteristic value for the pair (L, TV) provided that ker(TV + AL) 7^ {0} and we say that A is a regular covalue of (L,N) if the resolvent 7?(A) := (TV + AL)" 1 exists on X and it is continuous.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Since, from the above equivalence, we have
ker(TV + XL) = ker(/ - M(X))
(25)
it results that each A > 0 is a characteristic value or a regular covalue for
(L,N). Denote by C(L, TV) the set of characteristic values of (L, TV) and by La the spectral operator for (L, TV) La := (N + aL)~lL = R(a)L,
(26)
and observe that La is compact whenever L is compact. From the identity
(X-a)La]
(27)
we see that TV + XA is invertible if and only if a — A is not a characteristic value of La. Hence from Fredholm theory it results that C(L, TV) is at most a countable set with only one possible acumulation point at in0nity, as we have seen in the theorem 1. Suppose now that a is a 0xed covalue of (L, TV) and de0ne the multiplicity
m(A) of the characteristic value A 6 C(L, TV) to be the algebraic multiplicity of the eingenvalue (a — A)" 1 of the compact operator La. Let a G D be an isolated solution of the equations (21) and choose an open ball B(a, r) such that R(X)f fl B(a, r) = {a}. By the excision property of the coincidence degree we can de0ne the coincidence index of (L,N) in a with respect to / to be
ix((L, TV), a, /) := d x ( ( L , N),B(a, r), /)
(28)
and, by the above equivalence, it is true the following tranversality property: Proposition 1 (Leray-Shauder) // [Ai, A 2 ] f~l C(L, TV) = {A}, then
iXl ( ( L , TV), a, /) = (-l)">Wi Aa ((i, TV), a, /). In the semilinear case it apperas naturally the following question: Are the well known results of Krosnoselskii and Rabinowitz, concerning bifurcation theory for compact operators, true for monotone type operators? The answer is Yes! and was given in [5] in the following way: Let X be a re/Eexive uniformly convex Banach space and X* be its dual Banach space. As usually we denote by || • || the norm in both spaces, by < •, • > the duality pairing and by J : X —>• X* , the duality map. Consider the eingenvalue problem
Jx + fj,Ax + R(n,x) = 0,
(29)
where A : X —> X* is a linear continuous operator and R(/J,, •) : X —-*• X* is a nonlinear perturbation such that R(n,Q) = 0, V/u G K. Note that only J is monotone. In this case (//, 0) are solution of (29) for all n G M-named trivial solutions and the set of all trivial solutions are denoted by C.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
A point (/J.0,0) 6 C is said to be a bifurcation point for (29) provided that there exists solutions (/J,a^), x^ ^ 0 in each neinghbourhood of (n0_,0). Let us denote by So the set all of these nontrivial solutions and let 5 := So be its aderence in M x X.
The key step in our extension is the Browder-Ton theorem concerning the compact imbedding property for separable Banach spaces (e.g. D. Pascali and
S. Sburlan [7,p.302]):
Theorem (Browder-Ton). Let X be a separable rejEexive Banach space and let S be a countable subset of X. Then there exists a separable Hilbert space H and a compact one-to-one linear operator ^ : H —»• X such that S C tl>(H) and ijj(H) is dense in X. De0ne now the adjoint operator
<<{>u,v>=(u,il>v),Vu<EH, v&X*.
(30)
Then the operator L := —il>
tr(L) n (/i U 72) = , where I\ := (en0 — 6,e/J0) and 72 := (ep0,£fi0 + 6) and /j,0 G
, i=l,2 are regular and thus there exists M > 0 such that
— i a ; > M \\x\\, \/x<=X, i= 1,2.
(31)
Suppose that A is bounded from below in the sense
(i)
and the complementary part is asimptotical zero, i.e.,
uniformly in [i on bounded sets. Using the following Leray-Schauder homotopy
h t ( x ) := x + -ipipJx + -uibipAx, t G fO, 11, x e X. e
s
(32)
we easily obtain __ . I I 7* I p <* ^ A™ T> \<^ _ _ . l l - r l l ^ C ^ /T-ti* . iL s* ^ —
which is a contradiction, (C. Mortici and S. Sburlan [5]). By homotopy invariance of the Leray-Schauder degree we can conclude that / I \ ( \ \ dis 7 + -/uV"M, 5 , 0 = dis 7 + -^$(3 + p.A), 5 , 0 .
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Let now use the Berkovitz de0nition of degree for (5+) mappings (e.g. S. Sburlan and G. Moroanu [10]) ds(J + nA, B, 0) = dLS(I + -^M, B, 0)
(33)
and consider the mapping
Since L is a compact map and, by (33)
where m(/u) is the sum of algebraic multiplicity of all eingenvalues A > - of L. It then follows that f(n^) = — <^(// 2 ) whenever the eingenvalue AO = -^- has odd algebraic multiplicity because it appears only in the expresion of Therefore in this case
ds(J + KA, B, 0) ^ ds(J + n2A, B, 0).
(34)
Take now the Leray-Schauder homotopies H\ : [0, 1] x X -—> X, i — 1, 2 Hi(x) = (I-^L)x + -^(Jx + R(^,x)), t € [ 0 , l ] , x€X, £
C
and therefore,
dLS(I - ^L, B, 0) = dLS(I + i( J + ^A + R(^, •)), B, 0), » = 1,2. Combine this equalitty with (34) to produce
ds(J + ^A + R ( V I , - ) , B, 0) ^ ds(J + n2A + R(»2, •), B, 0). Join linearly fj,l with /K 2 by nt := (I — t)^l + t// 2
an
d consider the (5+)
homotopy
X t x ) := J x Hence it could exist T G [0, 1] and xr G dB such that JxT + nTAxT + R(nT , XT) = 0, that is the equation (29) has nontrivial solutions in any neighbourhood of (e//o> 0) G M x J5^, and thus (efiQ, 0) is a bifurcation point for (29). Thus we have proved an analogous of Krasnoselskii theorem for monotone type operators (e.g. S. Sburlan [8]). Theorem 2. (Krasnoselskii). Let /j,0 be a characteristic value with odd algebraic multiplicity of the linear compact operator L G L(X). If there exist £, 6 > 0 such that (i)-(ii) hold, then (£/i 0 ,0) G R G X is a bifurcation point for (29).
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Example 2. If consider R ( f i , x ) := / j , \ \ x \ \ 2 J x , then ,x),x >= H\\x\\2 < Jx,x >= H\\x\\4. The above results can be applied in Sobolev space X :— H1^). The corre-
sponding operator is — Aw + /i||u||Aw or p-laplacian operator in the general case X := Wl>p(tt). m Let us denote
i- :=ds(J +LiA,B,Q) = dLS(I + -^(j)A, B,0), fi G h,
i+ :=ds(J + nA,B,Q) = dLS(I+-^(t>A,B,Q), n 6 72, and observe that these degrees are constant in /i1 G /i and // 2 € hFor any 0xed r > 0 de0ne the mapping Hr : M+ x X —» M+ x X* as follows
# r (//, x) := (\\x\\2 - r2, Jx + nAx + R(n, x ) ) , V(//, x ) < E R x X and we have a formula similar to Ize's formula:
d.(Hr,B,Q) = i--i+,
(35)
where B = {(v, x) £ R x .\>2 + ||a;||2 < <52 + r2}.
Indeed, by de0nition of (S+) degree (e.g., S. Sburlan and G. Moroanu [10]) we have
where
Ur(fi, x) := (\\x\\2 - r2, (I - pL)x - N(n, x)), V(//, x) e M x X, and
, x ) ) = o(\\x\ We can now prove a global result concerning the bifurcation under monoto-
nocity condition similar to those under compactness condition proved by Rabinowitz. Theorem 3 (Rabinowitz). If S is a connected component containing the bifurcation point (e// 0 , 0) G S, then we have one of the following two posibilities: (j) £ is unbounded in M x X . (jj) £ contains a 0nite number of bifurcation points (e/^-,0) where ?t e "WMoreover, the number of these points, including (£/^ 0 ,0), is even.
Proof. Suppose that £ is bounded in M x X and let G be a, bounded domain that contain £ and a 0nite number of points (e^ 0 ,0). Moreover, suppose that there is no solution of (29) on 3D. In this case d(Hr,G,0) is well de0ned and independent of r > 0. For r enough small
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
where Gj are disjoint and each Gj contains only one point (s/j,j,0).
It then
follows from (35) that
where the sum has only an even nonvanishing terms because i*- (j) — J+ G {0, ±2}. If mj is the algebraic multiplicity of e^ij , then i + C/) = (-l) m ''-(j),
that is i-(j) ^ i + ( j ) , which is possible if and only if ep,j has an odd algebraic multiplicity. • We conclude our work with the following Example 3. Let g : I x M 2 —» M be a bounded continuous function
\g(t,t,ri)\<M, w e / , (£,??) e M 2 and consider the eingenvalue problem u"(t) + \u(t) + g(t, u(t), «'(<)) = f ( t ) , i € /
(£«)(*) = 0, i e <9/.
where J := [0, 1] C M, B denotes either Dirichlet boundary conditions u(0) = w(l) = 0 or Neumann boundary conditions
u'(0) = «'(l) = 0, or periodic boundary conditions
w(0) = «(1), u'(Q) = u'(l). Such Take < G $/} (N(u)(t)
problems was extensively studied in the last time (see e.g. Drbek [1]). X := L2(I), L : D(L) ->• X with D(L) := {u e C 2 (I); (5w)(i) = 0, de0ned by LM := —u", N(u) : X —> X the superposition operator := g(t,u(t),u'(t)) and / £ C(/). With these notations we arrive to the
following equation in X.
Lu + \u + N(u) = f with L G L ( X ) symmetric and positive. Since L is invertible with i
(Au)(t) = L~lu(t) := f G(t, s)u(s)ds, o
u £ X, t € [0, 1]
where G : [0, 1]2 —»• M is the continuous function
it results that A is compact and the above equation can be written as
u + XAu + N(u) = Af. Now we can apply the above results. •
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
REFERENCES [1] P. Drbek, Solvability and Bifurcations of Nonlinear Equations, Pitman Res. Not. Math., 264, Longman, London, 1992. [2] A. Haraux, Nonlinear Evolution Equations. Global Behaviour of Solutions,
Lect. Notes Math., Vol. 841, Springer - Verlag, Berlin, 1981. [3] G. Morosanu and S. Sburlan, Multiple Orthogonal Sequence Method and
Applications, An. St. Univ. jOvidiusj Constanta, 2 (1994), 188-200. [4] C. Mortici, Bifurcations for Semilinear Equations with Compact Nonlinearities, Bull. Appl. Com. Math., BAM-1714/'99XC-B, pp.265-272.
[5] C. Mortici and S. Sburlan, Bifurcations for Semilinear Equations of Monotone Type, An. Univ. Ovidius Constanta, vol. 7, (1998) fasc.2, pp. [6] J. Neas, Les Mthods Discretes en Thorie des quations Elliptiques, Ed. Academia, Praque, 1967. [7] D. Pascal! and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijhoce
& Noordhoce Int. Publ., 1978. [8] S. Sburlan, Gradul Topologic. Academiei, Bucureti, 1983.
Lecii asupra Ecuaiilor Neliniare, Ed.
[9] S. Sburlan, Luminita Barbu and C. Mortici, Ecuaii Difereniale, Integrate i Sisteme Dinamice, Ex Ponto, Constanta, 1999.
[10] S. Sburlan and G. Morosanu, Monotonocity Methods for Partial Dioeerential Equations, MB-11/PAMM, Budapest, 1999. [11] E. Zeidler, Applied Functional Analysis, Appl. Math. Sci., 108, Springer-
Verlag, Berlin, 1995.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Implied Volatility for American Options via Optimal Control and Fast Numerical Solutions of Obstacle Problems
Srdjan Stojanovic Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025 U.S.A. [email protected] http://math.uc.edu/~srdjan http://CFMLab.com December 2000
• 1. Statement of the Problem Let S(t) denote the price S of a particular stock at the time /. We suppose that the price evolves on the stock market according to the stochastic differential equation dS(t) = 5(0 (a(t, 5(0) dt + cr(t, 5(0) dB(t)}
where a(t, S) is the appreciation rate,
attention to American put options. Suppose the holder of the put option does not own the stock, and wishes to exercise at some time T. That means he/she buys the underlying stock on the open market at the price S(T) and then sells the same stock at the strike price k. The payoff for the holder is equal to k - S(r). Moreover, since option does not have to be exercised, the payoff is never negative - i.e., the payoff is equal to 0(S(r)) = Max(0, k - S(T)). According to the celebrated Black-Scholes theory [2] (treating European options), the fair price of an American put option with fixed strike price k, and fixed expiration time T, as a function of the current running time t and the current price of the underlying stock S, is the unique solution of the (degenerate) backward parabolic obstacle problem: 8i/i(t, S) + 1 s2 d <),(t S) ^ ^2 +/ . 5 d
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
2
(7O
OO
tf/(t, S) > Max(0, k-S)
together with the terminal condition i/i(T, S) = Max(0, k-S)
where r is the interest rate, Max(0, k - S) is the obstacle as well as the terminal condition, N is the non-coincidence set, T = dN n {(t, S);t
that if the price of the underlying stock drops below certain threshold (the free boundary T), then it does not make further sense to hold (or trade) the particular put option, and instead, such an option should be exercised.
On the other hand, according to the Dupire theory (see e.g. [3] for the case of European options), the fair price of an American put option with fixed underlying stock price S, at fixed time t, as a function of the expiration time T and the option strike price k, is the unique solution of the (degenerate) forward parabolic obstacle problem (written in the free boundary problem form):
V(T,k)>Max(0,k-S) V(T,k)= Max(0, k-S) onF = dNf] {(T, k); t0 < T, k > 0)
dV(T,k) , „ -1^ = 1 0 n r
together with the initial condition V(tt>, k) = Max(0, k-S)
The trading significance of the free boundary now is that for the given current price of the underlying stock S, only options with strikes below the free boundary should be considered for trading. It is remarkable that the same volatility function or, with different arguments, appears in both problems (as well as in the underlying stock price
evolution SDE above). Also notice that in either one of the problems the underlying stock price appreciation rate a(t, S) does not appear (but instead the known interest rate r does), while volatility cr is of the deciding importance (options with different underlying volatilities have significantly different prices). It is then of paramount importance to have a precise estimate of the volatility cr. Volatility of the underlying stock price cr can be estimated from the historical statistical data accurately. Nevertheless, the volatility changes, and in the above equations it is the future volatility that appears and not the current or past. Moreover, knowing or having a good estimate of the future volatility may be useful in trading for other
reasons, as well. Therefore it is of great importance in trading options or even stocks, to have a reliable and efficient technology for estimating future volatility cr based on the current prices of the whole variety of the corresponding call and put options. For example, for the underlying QQQ (Nasdaq-100 Index Tracking Stock), AS OF DEC 26, 2000 10:46:20 AM (E.T.), the collection of sufficiently liquid put options with expiration in January and
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
February of 2001 (T = 1.05593, 1.13259; time is measured in units of years, and we start at the year 2000), with different strike prices (the second column), had the following market prices (the third column): .05593 .05593 .05593 .05593 .05593 .05593 .05593 1.05593 1.05593
48 0.781251 50 1.03125 51 1.1875 52 1.375 1.625 53 54 1.875 55 2.1875 56 2.375 c 1.13259 48 57 2.75 1.13259 50 1.05593 58 3.125 1.13259 52
1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 1.05593 ,1.05593
59 60 61 62 63 64 65 66 69 70 71 75
83 86
3.6875 1.13259 4.3125 1.13259 4.6875 1.13259 5.1875 1.13259 5.8125 1.13259 6.375 1.13259 6.875 .1.13259 7.625 9.875 10.75 11.5 14.875 22.75 25.75.
55 56 59 60 65 75 80
1.6251
2.125 2.625 3.6875 4.0625 5.3125 5.8125 8.8125 15.8125 20.1875,
(Notice, by the way, how much more trading activity is performed for options that expire sooner rather then later.) The question is what can be said about the (market consensus about the) future volatility cr of the underlying stock price based on these recorded prices. The present paper describes a methodology for answering that question based on the optimal control theory for obstacle problems, as well as on a fast numerical solution of the Dupire obstacle problems. Some alternative methods can be found in [3].
• 2. Some Remarks on Obstacle Problems We recall some known facts about obstacle problems, introduce some new (see also [6,8]), as well as a method for solving obstacle problems, employed in our fast numerical solution of the Dupire obstacle problem.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
• 2.1. Various Formulations of Obstacle Problems Consider, for the sake of simplicity, an obstacle problem for the Laplace's operator and with zero obstacle. Everything that follows can be generalized to the case of arbitrary, sufficiently regular obstacle, as well as to the case of an arbitrary elliptic (or parabolic) operator with smooth coefficients. Obstacle problem can be formulated in many different ways. Arguably, the most popular way is the variational inequality formulation. Let fl c R" be a bounded domain with regular boundary, let (-/) e £2(fT) be the right hand side, and let g e //'(H), g > 0 be the boundary value. Denote also //g(H) = g + //o(fi), and K = (v e //g(H), v > 0}. The variational inequality formulation of such an obstacle problem is:
Find v e K, such that
for any ip e K. It is worthwhile emphasizing that the obstacle appears explicitly as a constrain! imposed on all
functions considered as possible solutions. As it is very well known, such a problem has an unique solution. The variational inequality formulation is useful because it allows an easy variational proof of existence of the weak solution, as well as its uniqueness. Furthermore, the higher regularity of the weak solution can be established posteriori via additional arguments. Indeed, under above assumptions v e 7/£c(fl) (see e.g. [5]). Increasing the regularity of the right hand side yields higher regularity of the solution, but not beyond ^O'"(fl) (unless obstacle does not affect the solution). The solution v,
being non-negative, determines two distinctive regions in ft: the coincidence set A = (v = 0) D Q and me non" coincidence set N = {v > 0) f| ft. The boundary separating the two T = d{u > 0( f| H is called free boundary. In
general, free boundary is not a smooth surface no matter how smooth is the right hand side (see [4] for fundamental results ensuring smoothness of the free boundary under additional assumptions; see also [5]).
• 2.1.2. Semi-Linear PDE Formulations of Obstacle Problems Here are couple, not so well known, formulations of the same problem, introduced and utilized by the author in [6,8].
Let IA(X) be a characteristic function of the set A, i.e., let IA(x) = 1{
1, x e A 0, x $ A
. Also let
h+ = Max(/z, 0), h~ = -Min(h, 0) be the positive and negative parts of h, so that h = h* - h~. The above obstacle problem can be formulated as: Find v e H*(£l) f| H^(fl) such that -Av + // (v>0 ) = 0 /" /(v=0| = 0
All the equalities are to be understood as equalities almost everywhere. The above obstacle problem can also be formulated as:
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Find maximal v e //j(ft) f| //,2c(ft) such that -Av + //,v>0) = 0
Alternatively, let
-W.* = fw e tf](n) 0 tfi2oc(«)> -A v + / /,,,>0) = 0) then v(x) = Max^A-,AU w(x).
X\fljg is typically a small set. All the elements of X\ fljg are non-negative functions due to the maximum principle, and if/ > 0 then X\fijg is a singleton consisting of the solution of the obstacle problem. Notice also that the non-negativity follows although no constraint is imposed explicitly in either of the above two formulations. For example, let ft = (0, 1), g= 1, f(x) = -87T2 cos(47rx). Then ^i,n,/,g has two elements: wiM = 7 (cos(4 n x) + 1) and w2 (x) = j (cos(4 nx) + 1) ^<±)U(jt>i)> me fifst one being larger, and therefore being
the solution of the obstacle problem. Notice also /" /(W2=o) = -8^ 2 cos(4wx)/ ( | < . t< ||W * 0.
• 2.1.3. Classical Free Boundary Problem Formulation Assume / is sufficiently regular. The free boundary problem can be formulated as: Find an open set N c ft and v e C2(AO f| C\N), such that v>0
and -Av+/=0
in N, such that v =g
on a N 0 3ft, and such that
and Vv = 0
on dN fl ft- The last two conditions are called free boundary conditions.
Notice that if / is sufficiently regular, any element of ^i,n,/,g is a solution of the free boundary problem. This implies in particular that the free boundary problem formulation does not yield necessarily an unique solution. In the above example, both w\ and W2 are solutions of the free boundary problem.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
• 2.2. Maximum Boundary Value Formulation of the Obstacle Problem For each open N c R", let Aw = R" - A'. Now let g,w | AV = 0}
Notice that w e H^£N(fl) has more then one extension, i.e., representation in //^(R"). Further, let WN be the unique function in //(jgA,(i"l) such that -AWJV +/ = 0
Notice that if N f~) fi = 0 then vf// = 0. On the other hand, in order to see non-trivial examples of WN, let n = (-l, 1), g = 1 ,/(*) = -10+ 20/(,>(» and let ty = (-co, }), jV2 = (-l, oo). Then t.~ .
•,
. •,
141x 28
21\ . 28^
which looks like
while w^OO = (10 /(r>0) x2 - 5 ^ - -^|i - ±)
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
, which looks like
It turns out that the pointwise maximum of functions such as these, yields the solution of the obstacle problem,
which by the way in this example is equal to V(JC) =
i
]+
rrn + (5*2 + 2 (-5 + ^
't«'-vr)
and looks like
-0.5
0.5
So, formalizing, let XHJJS = (WN, N c R")
Compared to X\fijf, Xfij^ is much larger. Also, one should notice that, like in the examples above, depending on N, as opposed to X\fijg the elements of X&jf take negative values, i.e., values below the obstacle, as well.
As announced, the obstacle problem can be formulated, and/or solved, simply as v(x) = Max.reA^ w(x).
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Before proving this claim, it is instructive to visualize this maximization (on the same example), together with the solution, and its free boundary:
The following simple facts may have practical implications for numerical searching of the free boundary
(v>0) = l I
\JweXnsj
(w>0)
or alternatively, (v = 0} =I n\wsXn. (w<0) u
and consequently, for any Y c Xn
{v > 0) D (J
(w > 0)
and
In particular, for any Y c Xnj-g, the set fXe)' {w s 0} contains the free boundary d{v > 0} (~) ^- In order to prove the above claim, since v e Xnjg, we need to show only jv > 0} D Uwe^n, f w > "'• ^et *e (w > 0} for some w e Xftj£. This implies v(x) > w(x) > 0, and therefore x e {v > 0}.
This formulation-solution of the obstacle problem can be compared - judged against the classical free boundary problem formulation. The free boundary problem can be viewed as the first order necessary condition for the above
maximization problem. The computational efficiency of the present method lies in the fact that for each N, computing the corresponding WH is a linear problem. Still, in general, especially in higher dimension, choosing proper A"s, i.e., choosing sufficiently representative finite family Y c ^n,/,g mav indeed be difficult, or at least additional issues will need to be addressed there. Nevertheless, in special cases such as the Dupire obstacle problem, when the
geometry of the solution and of the free boundary is well understood a priori (in spite of being unknown), the above maximization can be done very efficiently.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
• 2.3. A Proof of the Equivalence We shall prove that if v(x) = MaxweXn.fg w(x). and if z e H^(Cl) f) H?x(n) satisfies /(z>o) = 0
then v = z. Indeed, since z e An_/,£, all what is needed to prove is that if WN e A'n./.g, then z > w^. To that end recall I
Jnrw
for any ip e //0' (H f| AO, and f( Jn
for any y e H^(Cl). Consider (WN - z)+ e H^(fi). Notice also (ww - z)+ lnnAA. = 0. Therefore f (Vw w -V(w w -z) + +//A, (WAf -z) + )dx = 0 Jn
and f (Vz- V(w w -z) + +// (z>0 ) (wN -zDdx = 0
Jn
Subtracting 0= f(|V(w^-z) + |2 +(/ + -/-)(/ w -/ (r> o))(w A ,-z) + )^ = Jn
f ( I V(WN - z)+ |2 +/+ (1 - ;,z>0)) (WAT - z)+ - /" (/„ - 1) (WN - z)+) d^ > f ( | V(wN - z)+ |2) dx
Jn
Jn
since /" /(_->o) = /" /(:>oi = /" , yielding V(wN - z)+ = 0, and therefore (WN - z)+ = 0, and finally z > WN.
m 3. Fast Numerical Solution of the Dupire Obstacle Problem Back to the Dupire obstacle problem (free boundary problem formulation):
Y(T, k) > Max(0, k-S)
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
V(T, k) = Max(0, k-S) on T = <9 W fl ((T, k); ta
d V(T, k) • = 1 on F dk
together with the initial condition V(t0, k) = Max(0, k-S)
Under some theoretical assumptions, that we shall not be going into here, the Dupire obstacle problem admits an
unique solution. This problem is approximated by the time finite difference variant: • V(T, k)-V(T-dT,k)\ • 1 2, d2 V(T,' k)' urr(T -\+-k ~ , ~——— \* > V dT I 2 ok 6P20 inN = {(t, S); V(T, k) > Max(0, k-S)} v
]f\ >*)
,
d V(T, k) ok
— ~ tr K.I- ———~——— —
V(T, k) > Max(0, k-S) V(T,k)= Max(0, k-S) onT =
, k); ta < T, k > 0)
dV(T,k) = 1 onT dk
i.e., by the family of 1-dimensional elliptic obstacle problems, that are solved (numerically) in succession. For example, if
150 100 50
while, more precisely, its free boundary looks like
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Free Boundary 80
1.02 1 . 0 4 1 . 0 6 1.
The geometry of the solution is very simple: the free boundary is a graph of a function, i.e., T = {{T, g(T)}, {T, to, t\}}; moreover function g is strictly increasing (initiating at the price of QQQ AS OF DEC
26, 2000 10:46:20 AM (E.T.)) . Those two facts can be exploited in the search for the maximal boundary value solution of each 1-dimensional elliptic problem, like in 2.2.
• 4. Optimal Control Problem The above Dupire obstacle problem can be written more precisely, for example in its complementarity form d V(T, k} 1 2 d2 V(T, k) + k ——dT~ 2 dV
'
2
~'
y(T,k)>Max(0,k-S)
together with the same initial condition above. Also, for the purpose of numerical solutions, we need to restrict the consideration on a finite domain, say {(T, k);t$
dT dk
0, f ' f [(
For each
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
/do-(T, k)\2
,(-^) + , 2 «r(r,
where e\, £3, qi(k) > 0 are given and crslat is the statistical estimate of the constant volatility.
The optimal control problem is to find a-aft e K
such that •/(^opt) = Mm(J(cr), creK)
m 5. Derivation of the Minimization Algorithm The state equation being what it is, the functional J(
inequalities (see e.g. [1,7],). Nevertheless from the practical point of view, from the point of view of designing a working numerical optimization algorithm, this is not a problem at all. Indeed, for a fixed
J' (a-; ZT) = lira s_>0
v
s/
' = /> , i
(K(r/, k) - v,(sj) w(T,, k)q,(k)dk + /i/i. v
+
/o
h, \
dk
dk
I
where ,T k) M = Mm iw(T,
<5->0
S
is the unique solution of the forward equation dw(T,k)
+
I
k
62w(T,k)
——dT~ 2 —M-
a(T k)
,
> ~
rk
dw(T,k)
l2d
2
V(o-)(T,k)
,„„_,„„
—ST- = ~* ——dit —— ^ k)
in the non-coincidence set # = {(T, k); V(cr) (T,k)> Max(0, k - S)} [~] {(T, k); I0 < T < t\ , kg < k < k\ }, and w(T, k) = 0
in the coincidence set A = {(T, k); V(cr) (T, k) = Max(0, k - S)}. Let for any ! < ( ' < « , p,(T, k) be the unique solution of the backward equations (not obstacle problems) in the non-coincidence region:
(a-(T, kf +k(4 o-(OJ)(7", k) + k o-(0'2)(7", k)) cr(T, k) + k2 o-(0'[\T, k)2 + r) p,(T, k) = 0
for N 0 {(T, k); t0 < T < T,}, with the terminal condition Pi(Ti, k) = -(V(
and (non-cylindrical) boundary condition
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
on the lateral boundary of N f| {(T, k); t0
JJ
_____). + l
Nr\{(T,k);l,
-
r
Jka Jka
f'(
Ti, k) - Vi(k))qi(k)
Jka
for any smooth test function (/> such that 0 = 0 on the backward parabolic boundary of the non-coincidence set. In particular, if 0 = w,
Z JJ
JJ ff Jl0 e
Jka
aV(r, k)
\
I r" da-(T, k,)
U
U,0
.
e, ——-^— + to (a-(T, k) - o-5tat) v(T, k)dkdT +
——HT——cr(T,ki)dT-
dk
r"
j,0
dk f
On the other hand, there exists z such that
2
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
W7.
fPrrtT
k\
', k) - o-stat)
Since /'(cr;a°) = VJe(cr)-o° for any admissible a, we conclude that, for any T, z(T, •) is the solution of, and consequently can be computed as a unique solution of the boundary value problem for an ordinary differential equation (regularization equation): d2z(T, k) dk2
eo z(T, k) = -k2
-o-(T,k)}_i(Pi(T,k)X(,{,,n. /=i
ff-
H to (
with boundary conditions dz(T, kg)
=
dk
~
da-(T, kg)
dk
dz(T,k,)
=
da-(T,k,)
dk
~
dk
We summarize the steepest descent minimization algorithm. The single iterate is done in 4 steps: 1) for given cr(T, k), V(cr)(T, k) is computed as the unique solution of the Dupire obstacle problem;
2) pi(T, k), i = l , ..., n are computed using V(cf)(T, k) as solutions of the adjoint equations in the noncoincidence region; 3) V(
4) a-nextC?1, k) = a-(T, k) -pz(T, k), for some p > 0.
• 6. Example: Put Implied Volatility for QQQ (Nasdaq-100 Index Tracking Stock) We proceed with the data shown in Section 1. The first step is to construct the target functions v,(A), i = I, 2, that the solution of the Dupire pde is supposed to match. They look like v
il k l (Expiration) T = 1.05593 25
55
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
60
65
70
75
80
85
k (Strike)
V
2 [ k ] (Expiration) T = 1.13259 20
17.5 15 12.5 10 7.5 60
65
70
75
—— k ( S t r i k e )
The steepest descent method is initialized by selecting the volatility function to be a statistical volatility cr(T, k) = o-stat; see above. After a number of iterations, on two different grids, the optimal volatility function, as a function of strike and expiration time looks like
150 100 50
The corresponding solution of the Dupire obstacle problem, together with the free boundary, and together with the observed prices, looks like
1.1
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One can notice two features above. One is the reduced smoothness of the free boundary, as compared to the one presented before in the case of the constant volatility. The second feature is that the observed option market prices are all below the computed free boundary, as theory postulates: options with strikes above the free boundary need to be executed, and not be held or traded. Proceeding, the corresponding gradient (the smallness of the gradient is the measure of the success of the minimization procedure; more iterations on the same grid, as well as going down on the finer grid and performing a number of iterations there, would reduce the gradient, and the precision of
150
100 50
while the solution of the Dupire obstacle problem was matching the targets quite well: 25 20 15
55.
65
70
75
80
85
20
15
75
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
80
Finally, in order to concentrate our attention on time dependence of the implied volatility, we average the computed implied volatility for each time. The average, superimposed with points representing times of expiry (January and February 2001), looks like
It is interesting to compare the term structure of the put implied volatility, with the one computed for calls (instead of the Dupire obstacle problem, Dupire partial differential equation and it's optimal control was used) on the same underlying (the constant statistical volatility is plotted, as well). The comparison, i.e. the mutual relationship between the two may suggest some insight about the option traders short term outlook of the (Nasdaq) market: AS OF DEC 26,
2000
1 0 : 4 6 : 2 0 AM ( E . T . )
$60.25
(Indeed, if an option trader expects QQQ price to increase, he/she would like to buy a call option (sell a put option). If many option traders share that opinion, the price of the call will go up, and price of the put (on the same underlying) will go down, and consequently its implied volatilities, as well. So, the difference in implied volatilities for calls and puts on the same underlying can be attributed to such a scenario, and consequently, can be exploited.) Although the market outlook sometimes changes suddenly, later that same day, the outlook did not look much different,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
AS OF DEC 26,
2000
4 : 1 5 : 2 2 PM ( E . T . )
$60.88
Many more details, including the complete implementation of the above algorithm, can be found in [9].
• References 1. V. Barbu, Optimal Control of Variations! Inequalities, Pitman, 1984. 2. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ. 81 (1973), 637-659.
3.1. Bouchouev and V. Isakov, Uniqueness, Stability, and Numerical Methods for the Inverse Problem that Arises in Financial Markets, Inverse Problems, 15 (1999), R1-R22
4. L. A. Caffarelli, Regularity of free boundaries in higher dimensions, Acta Math. 139(1977), 155-184. 5. A. Friedman, Variational Principles and Free Boundary problems, Wiley, 1982.
6. S. Stojanovic, Remarks on W^-solutions of bilateral obstacle problems, IMA preprint #1318, University of Minnesota (1995).
7. S. Stojanovic, Perturbation formula for regular free boundaries in elliptic and parabolic obstacle problems, SIAM J. Control AOptimiz., 35 (1997), 2086-2100. 8. S. Stojanovic, Modeling and minimization of extinction in Volterra-Lotka type equations with free boundaries, J. Differential Equations 134 (1997), 320-342.
9. S. Stojanovic, Computational Financial Mathematics, (book and CD-ROM in preparation).
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
First Order Necessary Conditions of Optimality for Semilinear Optimal Control Problems M.D.
Voisei
Dept. of Math., Ohio University, Athens, OH, 45701, USA
Abstract This paper is concerned with the providing of necessary condition of optimality for optimal pairs (y*,u*) with respect to the cost functional g(y) + h(u) subject to Ay + Ly = Bu + /, where A, B are linear and L is Lipschitz continuous. One example of applicatios of our necessary condition to a semilinear elliptic equation is presented in terms of Clarke's generalized gradient.
1. Preliminaries
In this article we investigate first order necessary conditions of optimality for optimal pairs of problem (P) Minimize
g(y)
+ h(u)
on all (y,u) G V x U subject to state equation f. Here (U, (•, •){/), (H, (•, •)# are Hilbert spaces, V is a Hilbert space which is not identified with its dual V* such that V C H C V* algebraically and topologically with dense inclusions, V is compactly imbedded in H,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
g : H —> IR and h : U —> 1R U {00} are given functions, L : H —» H is Lipschitz continuous, A 6 L(V, V*), B 6 L(C7, #) and f € H. We denote by • || the norm in V, with {•, •} the pairing between V and V*, with | • * the norm in V*, with | • H the norm in H and with | • \u the norm in U. Assume (P) has an optimal pair (y*,u*) £ V x U. The main idea is to linearize the state equation by introducing a new control variable and by penalizing the cost functional with an appropriate function we construct a sequence of approximative problems depending on (y*,u*) and a parameter e > 0, given by
(Pe) Minimize
g(y] + h(u) + ±\v-Ly2H + ±\u-u*2u + ±\y- y*
2 Hi
on all (y, u, v) e V x U x H subject to state equation f.
Under some suitable assumptions on g, h, B, A and L problem (Pe) has at least one solution. We will derive the approximative necessary conditions of optimality and prove that these conditions converge in some sense, for e —>• 0, to the necessary optimality condition of (y*,u*). Previous treatments of this problem for L = d
F°(x-v) =limsup (l/t)(F(x + tv)-F(x)}, x,v e X.
(I)
The Clarke generalized gradient dF : X —> 2X* of F is defined by
y € dF(x) iff (y,w)x*xx < F°(x;w), Vw e X,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(2)
where X* is the dual of X and (-.-}x*xx denotes the dual product between X and X*. When (X, (-, -)x) is a separable Hilbert space, we can construct the regularizations of F given by
)= /
J TRn
F(Pnx - \Anr)pn(T)dr, x 6 X,
(3)
where A > 0, n = [A"1], {em}m=i ig an orthonormal basis in X, Xn is the finite dimensional space generated by (em}^=1, Pn is the projection of X n
into Xre, that is, Pnx = £ (x, em)xem, An : Kra —> Xn is denned by A n (r) = m=l
£ r m e m , r = (n, ..,
m=l
> 0, />^ = 1, Pn(9) = Pn(-6), W e M ". We recall that FA is Frechet differentiable and if we denote its Frechet derivative with VF\, then the sequence (V-F\) is uniformly bounded on bounded subsets of X, lim F\(x) = F(x), Vx 6 X, and if x\ —>• x strongly in X and VF A (x A ) -> ^ weakly in X*, then ^ e 5F(x) (see Barbu [2]). For a proper, convex, and lower semicontinuous function h : U —> IR U {00} the directional derivative of h is given by Pn
h'(u;u) =lim (l/t)(/i(w + tu) - h(u)), u,ueU
(4)
and the subdifferential dh : D(dh) C U —> 2U is denned by 77 e 9/i(w) iff (77, u; - u)v + h(u) < h(w), Vw e U.
(5)
We refer to Motreanu & Pavel [6] for results of convex analysis used in this paper. We have considered the function SL : H x H —> IR, defined by
SL(y,v) = \\v - Ly\\ y,v £ H.
(6)
In the following Lemma we characterize its generalized gradient. Lemma 1.1 Assume L : H —» H is Lipschitz continuous. For z € H we define /z : tf -»• IR by
,x£H.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(7)
Then the following statements are true i) SL is locally Lipschitz continuous in H x H and fz is Lipschitz in H for every z G H,
ii) SLO(y,v;y,v) = (v - Ly,v)H + fQ(v_Ly)(y;y}, Vy,v,y,v e H, iii) (w, z) 6 dSL(y, v) iff z = u — Ly and u> e dfz(y), iv) H H ^ t f k / f . V K ^ e a S ^ y , * ; ) , where K > 0 is the Lipschitz constant of L. v) If (wn,zn) 6 dSL(yn,vn), n>l and
(8)
wn —* w weakly in .fiT, zn —> 2: weakly in /f, un —> v weakly in H, and yn -> y strongly in H,
then (w,z) E dSL(y,v). Proof. Easy calculations show that SL °(y, v; y, v) = _hmsup
(y,v)->(y,v),tlO
±(\v + tv - L(y + ty)\2H - \v - Ly\2H) =
=hmsup (v -Ly,v- \(L(y + ty) - Ly))H = (v - Ly,v)H + ffv_Ly)(y,y), y-*y,tlo
for every y, v,y, v G H, so, ii) is true and it implies iii). iv) For each (w, z) £ dSL(y, v) we have z = v — Ly and w 6 dfz(y]
that is
(w,y)H
< K\Z\H, where K > 0 is the Lipschitz constant of L.
v) We have zn — vn — Lyn and wn € dfZn(yn}. potheses imply
Therefore the given hy-
zn —> z = v — Ly weakly in H. From
(wn,v)H
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we infer that for arbitrary fixed v
< n>°
nand
(w n , v)H < -£(*«, L(yn + tnv) - Lyn)H + £, Vn > 1.
(9)
We can assume, by Mazur Theorem, that wn = V] Oilnwni —> w strongly in H and zn = Y^ a^zTli —> z strongly in H
(Here /n is a finite set of positive integers, a\ > 0 and^ o^ = 1). ie/n
Then (9) can be written as
(£n,v)H < -±(zn,L(yn + tnv) - Lyn)H + i, Vn > 1.
(10)
If we let n —>• oo in (10) then we get
=limsup -f (2, L(yn + * n u) - Lyjn < n—xx
because yn —> y strongly in .ff. Since f is arbitrarily chosen we conclude that w £ fz(y)
55 L (y,t;).B
and (w,z) G
Remark 1.1 Let us notice that in the previous Lemma we have proved the following w
n G Jzn (yn)
wn —>• w weakly in H, zn —s- z weakly in H, 2/n -»• y strongly in H, ,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
2. The Main Result
Let us consider the assumptions (Ai) V,U,H are Hilbert spaces, H is separable, V is dense in H and the inclusion of V into H is compact, (A2) g is locally Lipschitz continuous in H and g(y) > —C\y\n + D, Vy e H, where C > 0, D 6 H, (A3) /i : U —> IR U {00} is proper convex and lower semi continuous, (A5) L : H ^ H is Lipschitz continuous, that is
\Lyi - Ly2\ < K\yi - y2\H, VVl,y2 e H (K > 0),
(12)
(A6) A € L(V, V*) satisfies
(Av,v)>u\\v\\2,VveV
(u>0),
(13)
(Ay) A dominates L, i.e.,
uj\i > K, \\v\\2 where A! = infi11-^-; z; e V. v = 0). -71 M v
l/f
In the following we suppose that (Ai— A 7 ) are fulfilled. Theorem 2.1 Let (j/*,u*) £ V x C/ be a solution of (P). Then there exist p G V, £ G # such that
-A> - £ e ^(y*), B*p e dh(u*}, £ e 0/_p(y*). Proof. Let us prove first that (Pe) has at least one optimal solution (ye, ue, ve}. We define
AH : D(AH) = {y£V;Ay£H}cH->HbyAHy
= Ay,y£ D(AH)
and, since A is a linear homeomorphism between V and V*, we can consider D(AH) as a Hilbert space endowed with the inner product
,Vu,v G D(AH}.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Then ( D ( A H ) , (-, •}D(AH)} is compactly embedded in V and A G L(D(AH),H).
The above assumption imply inf(P £ ) > — oo and if (yn^univn} is a minimizing sequence, that is Ayn + vn = Bun + f and
inf(P e ) < g(yn) + h(un] + ± vn - Lyn 2H + \\un - u*fv + \\yn- y*\2H < < mf(P e ) + I
(14)
then (yn,un,Vn) is bounded in H x U x H. The state equation Ayn+vn = Bun+f shows, in fact, that (yn) is bounded in D(AH) so, we can suppose that on a subsequence (denoted in the same way for simplicity) we have yn -»• y strongly in V, v n ~^ v weakly in H, un —> u weakly in U, and Ay + v = Bu + f . If we let n —> oo in (14) we infer that (y, u, v) G D(AH) x U x H =: V is a solution of (P£). We denote this solution by (y£, u£, ve). Next, we show that (y£, ue, v£) -> (y*, u*, Ly*) strongly in V
(15)
Since (y£,u£,v£) is optimal for (Pe) we have g(ye) + h(u£) + ±\v£ -Ly£2H + \\ue - u*\2v + \\y£ - y*\2H < (16)
Hence, (ys,u£,v£] is bounded in V and on a subsequence it satisfies (y£,u£,v£) -» (y,u,v) weakly in V, y£^y strongly in V, h(u£) —»liminf h(ue) > h(u),
and Ay + v = Bu + /. By passing to limit in (16) we get
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
h(u)
(17)
limsup j-\v£ - Ly£\< inf(P) - g(y) - h(u) < oo,
(18)
lim v£ - Ly£\H = Q,v = Ly,
(19)
ej.0
ej.0
ej.0
g(y) + h(u] + \ limsup [\u£ - u*\l + \\y£ - y* ZH} < inf(P) <
<eg(y) + h(u),
(20)
because Ay + Ly = Eu + /. This final inequality proves that us —>• u* strongly in U, y£ —> y* strongly in LT.
But (19) implies w£ —> Ly* strongly in H, since Lys —> Ly* strongly in Lf. Therefore + f- Ly£ -^ Bu* + f - Ly* = Ay* strongly in H, i.e. y£ ->• y* strongly in D(A H ), so (15) is proved. For A > 0, we define
Ix(y, u, v) = gx(y] + h(u] + \SL(y, v) + \\u - u*\l + \\y - y*2H+
where g\ is the regularization of g in the separable Hilbert space H , given by (3). Using a similar argument as above, for every A > 0 problem
(R\) Minimize
on all (y, u, v) e V x U x H subject to Ay + v = Bu + /,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
admits a solution (y\, u\,v\) 6 V and we prove that (on a subsequence) (?/A, U A , vx) -» (ye,ue, v£) strongly in V x U x V* (21) Indeed, since Aye + v£ = Bu£ + f and (yx,u\,v\) is optimal for (^A), we have
M(RX) = I\(y\,ui,vx) < Ix(y£,u£,vs) < C£ < oo
(22)
and this provides us with the boundedness of (yx,u\, v\) in V. Hence, on a subsequence we can assume that
(yx, ux, vx) -+(y, u, v) weakly in V y\ —> y strongly in V, v\ —>• v strongly in V* , Ay + v = Bu + f , h(ux) -^liminf h(ux) > h(u) AJ,U
^) > SL(y,v}.
Then, by passing to limsup in (22) we get
+| limsup {\ux - u*\l; + \yx - y*\*H + \ux - u£\2H + \yx - y£\2H} < AJ.O
L
l°
< g(y) + A(S) + ^S (y,v) + lu-u*l + ±\y-y*\2H, and so
l i m s u p { u A - M £ ^ + \y\-ye2H} < 0. A|0
(23)
This implies UA —>• u£ = u strongly in U, y = y£, v = v£ and Ayx —» Ay£ = Ay weakly in H since v\ —> we weakly in ff. Hence yx —> ye weakly in _D(A# ) and eventually on a subsequence yA —>• y£ strongly in V, thereby (21) is proved. The necessary condition of optimality for (y\,u\,vx) can be easily deducted according to the principle that if F : X —> IR is a given function denned on a Banach space X and F ( z ) =inf F(x) then F°(z,v] > 0 for every v e X. In our case, for problem (R\) we find
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(Vgx(yx),y)H + hf(ux,u) + l(S^ + (2ux -u*- u£,u)u + (2yx - y* - ye,y)H > 0
(24)
for every (y, u, v) G V x U x H such that Ay + v = Bu, where
dSL(yx,vx] C H x H . According to Lemma 1.1, we have S^(yx, v\) = vx — Lyx,
(yx,vx} e df(vx-Lyx)(yx)
and S$(yx,vx)\H <
Relation (24) can be equivalently written as
vx) + 2yx - y* - ye,y}H - \(vx - Lyx,Ay)H > 0,
(25)
for every y £ D(An), and
h'(ux,u] + (±B*(vx - Lyx) + 2ux - u* - u£,u)u > 0, Vu e U, or,
dh(ux) 3 B*px - 2ux + u* + u£,
(26)
where p\ = -^(vx- Lyx) e H. We have
P\ -> PE •= -~(v£ ~ Ly£) weakly in H, Sy(yx, vx) -> w£ weakly in H, and since Sy(yx,vx) e df(Vx_Lyx)(yx)
then by Remark 1.1 we know that
If we let A -> 0 in (25), (26) then we get
(T£ + \w£ + y£- y*,y)H - \(v£ - Ly£, Ay)H > 0, Vy G D(AH),
(27)
where r£ e dg(ye), and
dh(ue) 3 B*p£ + u* -u£. Let ?]£ 6 D(AH} = {y <EV\ A*y e H} be such that
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(28)
A*r]e = r£ + ^w£ + ys-y* Then relation (27) can be stated as
(A^£,y}H = -(p£,Ay)H, \/y 6 D(AH)
(29)
But (A*ri£,y)H = (A*rj£,y} = (r]£,Ay) = (r]£,Ay)H. Therefore (29) is equivalent to rj£ = —p£. In this way we proved that p£ G D(A*H) and
-A*pe - \ws + y*-y££ dg(y£).
(30)
If we multiply the last relation with — pe then we get, according to (12), that U\\Pe\
2
-
Pe||(d
+ 1^1*) < 0, (Ci < 00),
since {Udg(ye}} is bounded in H. Hence
wv^Tbe H < u\\Pe\\ < \±we\* + Cl<^ We have uj\i > K, therefore, (pe) and (~w£) are bounded in H so, eventually on a subsequence we infer that p£ —> p weakly in H, -w£ —> I weakly in H. We can pass to limit in (29) and (30) because A* is weakly-strongly closed in H and B* e L(H, U) to get
-A*P-eedg(y*),B*P£dh(u*).
(31)
In order to determine a relation between p and I we start recalling that
w£ edf(Ve-LyE)(y£),
i.e.,
(we,h)H
y->2/e,*io and so for arbitrary but fixed h E H we have
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(±w£,h)H
(~£w£, h)H < £(p e , L(y'e + tsh] - Ly'£) + e. Because ^w£ =: l£ —> i weakly in H and p£ —> p weakly in H we may assume by the Mazur theorem that on a convex combination we have
£n = E c>44; -» £ strongly in H and pn = E a£pei —> p strongly in # •Je/n
J6/71
l
(Here Ira is a finite set of positive integers, a n > 0 and E
a
n = !)•
This allows us to find (£n, h}n < j~(pn, L(y£+t£h) — Ly£)H+£, and passing to limit with e —> 0 (n —>• oo) we obtain (£, /i) H
We investigate the following particular case of (P) (Pi) Minimize
h(u), on all (y,u) 6 HQ(&.) x [/, subject to (SE}
{ -Ay + P(y) = Bu + f
\y = 0
in ft,
in ft,
where ft is a bounded open domain of class C2 in IRn, V = -ffo(ft), f/ is a given Hilbert space, H = I/ 2 (ft), g,h,B,f satisfy hypotheses (A2), (A 3 ), (A4) and (3 : IR -^ 1R is a Lipschitz continuous function, i.e.,
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
P(r2-)\ < K\ri~r2 , Vr a ,r 2 e 1R (K > 0).
(32)
We may see (SE) as Ay+Ly = Bu+f, where A = -A : H%(ty and L : I/ 2 (f2) —> L2(J1) is given by
Theorem 3.1 Suppose, in addition to the above hypotheses, that (y*, M*) is a solution of (Pi) and
\i>K
(33)
where A = inf{|Vi>|i,2(£))/|i>| L 2(Q); v = 0} is the first eigenvalue of —A in Q. Then there exist p e H^ft), I e L 2 (O) such that (ii) B*p e (iii) £(x) e p(x)d0(y*(x))
a.e. x e fi.
Proof. Since all the hypotheses are fulfilled, we may apply Theorem 2.1 to find p e #£(n), ^ G L 2 (ft) such that
+ 1 3 -^*P, 5*p € dh(u*), The final relation is equivalent to
(t, h}H
J
and it can be described as I G dG(y*) where G : L 2 (fi) —> IR is defined by
Thus i(x) e p(x)d(3(y*(x)) a.e. x G O (see Clarke [4] or loffe and Levin [5]). Since —A* = A, the proof is complete. •
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
References
1. Aizicovici, S., Motreanu D. & Pavel N.H., Nonlinear Programming Problems Associated with Closed Range Operators, Appl. Math. Optim., Vol. 40, pp. 211-228, 1999,
2. BARBU, V., Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993,
3. BARBU, V., Partial Differential Equations and Boundary Value Problems, Kluwer Academic Publishers, Dordrecht, Boston, 1998, 4. CLARKE, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New-York, 1983,
5. IOFFE, A. D. and LEVIN, V. I., Subdifferential of Convex Functions, Trudy Moskov. Mat. Obshch. (translated in Transaction of the Moscow Mathematical Society), Vol. 26, pp. 3-13, 1972,
6. Motreanu D., Pavel N.H., Tangency, Flow Invariance for Differential Equations and Optimization Problems, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 219, Marcel Dekker, New-York Basel, 1999, 7. TIBA, D., Lectures on the Optimal Control of Elliptic Equations, The 5th International Summer School Jyvaskyla, Lecture Notes 32, Finland, 1995.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Lyapunov Equation and the Stability of Nonautonomous Evolution Equations in Hilbert Spaces Quoc-Phong Vu Department of Mathematics Ohio University
Athens, OH 45701, U.S.A. and Siu Pang Yung Department of Mathematics University of Hong Kong Hong Kong
Abstract. We apply the method of Lyapunov equations A*P + PA = —I to the question of exponential stability of the differential equation u'(t] = A(t)u(t),t > 0, in a Hilbert space E. Under some suitable conditions, we show that the solutions are exponentially stable provided that A(t) generate exponentially stable semigroups with exponential types < a < 0, and are slowly varying in some sense. Estimates are also given for the rates of convergence of the solutions to zero.
1. Introduction
Let A(t) be closed linear operators on a Banach space E such that the differential equation
u'(t) = A(t)u(t),t>0,
(1)
is well-posed, i.e., for each XQ from a dense subset T> C E, Eq.(l) has a unique solution
with u(0) = x 0 , which depends continuously on the initial value x0. In this paper, we are concerned with conditions which imply that solutions to Eq.(l) are exponentially stable, i.e. \\u(t)\\ < Ne-fft\\u(Q)\\ for every solution u ( t ) of Eq.(l), where N and a are positive constants. Note that if dim E < oo and Eq.(l) is autonomous, i.e. A(t) = A are independent of t, then its solutions are exponentially stable if and only if all eigenvalues of A have
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
negative real parts, i.e., there exists a positive number a such that
tr(A) C {A € C : Re A < -
. / — 1 —5 \ ,,,.,. / cost s'mt °=( 0 - l j ' ^ = ( - s i n t cost
A
Then
a(A(t}} = {-!}, for all t > 0, but the solution matrix is
_ ( e*(cos ^ + | sin t) e~3t(cos t - \ sin t) ' ~ { e'fsin t - | cos t) e-3t(sm t + f cos t)
m (
and therefore is unstable (see [2]). Various conditions have been obtained for equations (1) in a finite dimensional space £, that, together with the condition of uniform exponential stability of the matrices A(i),i.e.
||e sA(( >|| < Ne'", ((7 >0, N > I are independent of <),
(2)
imply the exponential stability of solutions of Eq.(l) (see [1,3,5,7,10]). All these conditions express the property that A(i] are slowly varying matrices in a suitable sense. To our knowledge, analogous results have not been obtained for Eq.(l) in an infinite dimensional space. In this paper, we consider the problem of exponential stability of Eq.(l) in infinite dimensional Hilbert space, and we extend some results of the above mentioned papers to this case. Namely, we prove that if A(i) are uniformly bounded (i.e.
sup t>0 ||A(i)|| < oo) and slowly varying in an appropriate sense, and if (2) holds, then solutions to Eq.(l) are exponentially stable. We also obtain estimates of the rate of convergence of the solutions u(i) to zero (Theorems 7 and 8). The method we use to obtain these results is based on the Lyapunov's equation A*(i)P + PA(i] = —I with the variable operators A(t). This method also allows us to extend the results to some cases involving unbounded operator coefficients. For other recent applications of Lyapunov equations in the stability theory of evolution equations in infinite dimensional spaces see [13-15]. Throughout the paper T>(A) denotes the domain of an operator A, and a (A] denotes the spectrum of A.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
2. Main Results. Assume that A(i] are closed linear operators on a Hilbert space E which satisfy the following conditions:
HI. A(t) = iA0 + Ai(t), where A0 : E —» E is a self-adjoint, generally unbounded, operator, A\(t] are bounded, and A0 commutes with Ai(t) for all i, i.e. from x 6
V(Ao) we have Ai(t}x e T>(A0] and AQAi(t}x = Ai(t)A0x, for all t. H2. sup^o ||Ai(i)|| < oo, and ||e^i(0|| < jVe~ CTS , (a > 0,7V > 1 are independent of t). H3. For every x G T) := "D(Ao), the function t H-> Aj(i)a; is differentiable and A\(t]x is a bounded operator from T> to E1, so that it can be extended by continuity to a bounded operator on E. Note that conditions H1-H3 include the case of Eq.(l) with bounded operators A(t), in an infinite dimensional Hilbert space (set A0 = 0). It also follows from the theory of evolution equations (see e.g. [8], Chapter 5, Theorem 2.3 and 3.1) that under these conditions there exists an evolution system U(t, s)(0 < s < t < oo) associated with Eq.(l), such that solution of Eq.(l) with the initial value u(0) = x is given by u(t) - U(t,Q)x,t > 0. Our method of investigation of the asymptotic behavior of Eq.(l) is based on the operator equation of the following form
PA + BP = C,
(3)
where A and B are closed linear operators on E and C : E —> E is a bounded
linear operator. A bounded linear operator P : E —> E is called solution of Eq.(3) if PV(A) C V(B] and P Ax + BPx = Cx for all x € T>(A). The following proposition is well known (see, e.g. [9], [12]). Proposition 1. Let A and B be generators of exponentially stable Co-semigroups {esA}s>o and {e5B}s>0, respectively. Then the integral fOO
P =- I
Jo
esBCesAds
converges in the uniform operator topology and is the unique solution of Eq.(2). Now let A(i] satisfy conditions H1-H3. We will need the following lemma.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(4)
Lemma 2. An operator P(t) is a solution to equation
P(t)A(t) + A*(t)P(t) = -I
(5)
if and only if it is a solution of
-I.
(6)
Proof. Assume that P(t) is a solution to (5). By Proposition 1, and since AQ and commutes with Ai(t) as well as with Ai(t)*, we have fOO
P
^
=
fOO
e
~Jo
6
'
l
S
~~Jo
6
6
S
'
hence P(t) is a solution to (6). The converse is proved analogously, d
Thus, for each t > 0,
fp(i\ ( t ) .— .—
/•oo
I
Jo
fsA"i(ty
e
j esAi(t) as
fj\
(I)
is a bounded solution of Eqs(5)-(6). Proposition 3. There exist a, (3 > 0, independent of t, such that
/?IN| 2 < (P(t)x,x) < a\\x\\\ for all x e E.
(8)
Moreover, one can choose a = -7= and fl = -^.
Proof. From (7) and condition H2 it follows that I I P1rt//•*' W I I 2_^~ < I/ j -f
/•oo
Jo
/ V 2Cp ~ 2 < 7 UO V s 1l1 lkOr l1 1l 2 —— — Ct n - 2 l l"^r l l 2 i
IV
where a = -j=. Hence {P(i)x,x} < a||x||2. On the other hand /•oo
'
J
o
yoo
/•oo
jf
~~ J o e-
2sM
2
1
~ 2
2
^||x|| = ^||a:|| = /?||x|| . d
Proposition 4. The operator function P(t] is differentiate and satisfies
||P'(i)|| < K\\A((t)l where K =
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Proof. That P(t)x is differentiate for every x G E follows from (7). Since
P(t)Ai(t)x + Ai(t)P(t)x = -x, it follows that
x + AWP'Wx = -[P(t)A((t)x + A'1(tyP(t)x]. Since by H2 A((t) is bounded, and generates an exponentially stable semigroup, we have by Proposition 1 /*OO
P'(t)x= \ Jo
esA^[-P(t}A((t}-A\(tYP(t)}esA^xds.
Therefore, from (8) we have /•oo
\\P'(t)\\ < JQ where K
= *£ = _|L .
JV 2 e- 2 <"[2a||Ai(i)||]
Box
Consider the Cauchy problem
u'(t}=
A(t)u(t)
« ( 0 ) = x0
(y>
where the operators A(i) satisfy conditions H1-H3. We need the following simple lemma. Lemma 5. Let (p(t) and f ( t ) be real functions on [0, oo) such that ip(t) > 0 for all t > 0, is differentiate, satisfies y'(t) < f ( t ) for all t > 0. Then
Proof. The required inequality follows from
Jo ds
~ Jo
Proposition 6. Let u(t) be a solution of Eq.(9) and P(i) be solutions of Eqs.(5)-(6). Then the following estimate holds.
(P(t)u(t),u(t))<exp{-\--2MK I La
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
f K(s)||dsl } (P(Q)x0,x0}, for all t > 0. Jo J) (10)
Proof.
Since Eq.(l) is well posed, if u(io)
=
0 for some to then u ( t ) = 0 for all
t > 0. Hence, we can assume that u(i] ^ 0 for all t > 0. By Proposition 3, (P(t)u(t},u(t}} > 0 for all t > 0. We have
= (P'(t)u(t),u(t))
+ (P(t)u'(t},u(t))
+ (P(t)u(t),u'(t)).
On the other hand,
(P(t)A(t)u(t),u(t)) + (P(t)u(t),A(t)u(t)) (P(t}A(t)u(t),u(t}} + (A*(t)P(t)u(t),u(t))
= =
-(u(t),u(t))<-(P(t)u(t),u(t}). Moreover, from Propositions 3-4 it follows
= 2MK\\A'1(t)\\(P(t)u(t),u(t)). Therefore
d f 11 — ( P ( t ) u ( t ) , u ( t } ) < \2MK\\ A'^H - - ( P ( t ) u ( t ) , u ( t ) ) , and (10) follows from Lemma 5. d We remark that Proposition 6 gives better estimates than those in [4]. From Propositions 3 and 6 we obtain the following result. Theorem 7. Let u(i) be a solution of Eq.(9). Then
IKOH 2 < 2aMexp f- f- - 2MK f \\A!,(s]\\ds\ } \\x0\\\ I La Jo JJ where a and K are constants defined in Propositions 3-4. From Theorem 7 we obtain the following stability result which are extensions of results in [1-5], [7-10]. Theorem 8. Assume that A(t) satisfy conditions H1-H3. In addition, assume that Ai(t) satisfy one of the following conditions:
7 : = - s u p | | A i ( S ) | | < — — | — — , or ZaMK s >o
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
(11)
Then Eq.(l) is exponentially stable, i.e. ||u(t)|| < Le~ u/ '||ti(0)|| for every solution u ( t ) and some positive constants L,u. Proof. If (11) holds, then HOI! 2 < 2aMe-ia \\XQ\\\ where a = - - 2MKf a
> 0.
Assume now that (12) holds. Choose £ > 0 such that
1aM By (12), there exists to such that, for every t > to, we have
ff HA'^Uds <et. Jo Therefore,
\\u(t}\\2 < 2aMe-ta, for all t > t0. D
We remark that the conclusions in Theorems 7 and 8 remain valid for mild solutions, i.e., for functions u(i] = t/(i,0)x, thanks to the well-posedness of Eq.(l) in the considered situation. In other words, there exist positive constants L, a such that \\U(t,s)\\
°
< oo
(13)
implies that (12) holds. On the other hand, if (13) holds, then, for every x G £>, the limit lim^oo Ai(i]x exists. In fact, ft
yoo
\\mAi(t)x= lim / A((s)xds + A^x = /
t-^ca
t-Kx JQ
JQ
A',(s)xds + Ai(0)x, for all x <5 V.
Hence A(i)x converges as t —>• oo, for all x € T>. However, even in this case Theorem 8.1 of [8, p. 173] is not applicable since A(t) do not generate, in general, analytic semigroups.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Examples. 1. Let E := L 2 (IR) and b(t) be a bounded differentiable scalar valued function such that b(t) < — a for some a > 0, and assume that one of the following conditions is fulfilled:
(a) sup t > 0 \b'(t)\ is sufficiently small; (b)f~\V(t)\dt«X>;
Define operators A(t) on E by
d2u A(t}u(x) := i—— + b ( t ) u ( x ) , u ( t ) € E. Then, as is easily seen, A(t) satisfy conditions H1-H3 with AO := J^j b(t)u. Consequently, solutions of the equation
an
d Ai(t}u : =
d2 ut(t,x) = i——u(t,x) + b(t)u(t,x),t > 0,x € IR, (J JL
converge to zero exponentially (in L 2 (IR)-norm). 2. We generalize the above example by putting A0 := A, where A is a Laplace operator on E := L2(1R3), or E := L2(Q) where fi is a bounded domain in IR3 with smooth boundary, and A is defined with an appropriate boundary condition. It is well known that AO is a self-adjoint operator (see e.g. [6, Chapter 5]). Furthermore, let Ai(t) be a uniformly bounded family of operators on E such that
|| e ^i(<)|| < Ne-"s, (a>Q,N >1 are independent of t ) , and assume that A\(i) satisfies (11) or (12). Then solutions to the equation
ut(t, x) = iAu(<, x ) + A^(t}u(t, x), t > 0, are exponentially stable.
References
1. L. Cesari, Un nouvo criteria di stabilita per soluzioni delle equazioni differenziali lineari, Ann. Scoula Norm. Sup. Pisa (2)9 (1940), 163-186. 2. W.A. Copell, Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629, Springer, Berlin, 1978.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
3. J.K. Hale and A.P. Stokes, Conditions for the stability of nonautonomous differential equations, J. Math. Anal. Appl. 3 (1961), 50-69.
4. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes
in Mathematics, vol. 840, Springer, Berlin, 1981. 5. C.S. Kahane, On the stability of solutions of linear differential systems with slowly varying coefficients, Czechoslovak Math. J. 42 (117) (1992), 715-726. 6. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. 7. L. Markus and H. Yamabe, Global stability criteria for differential Osaka Math. J. 12 (1960), 305-317.
systems,
8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 9. C.R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, Berlin, 1967. 10. G. Sansone and R. Conti, Nonlinear Differential York, 1964.
Equations, McMillan, New
11. H. Tanabe, Evolution Equations, Pitman, London, 1979. 12. Vu Quoc Phong, The operator equation AX—XB = C with unbounded operators A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567-588. 13. Vu Quoc Phong, On the exponential stability and dichotomy of Co-semigroups, Studia Math. 132 (1999), 141-149.
14. Vu Quoc Phong and E. Schiller, the operator equation AX — XB = C, admissibility, and asymptotic behavior of differential equations, J. Differential Equations 145 (1998), 394-419. 15. E. Schiiler and Vu Quoc Phong, The operator equation AX — XV2 = —<50 and second order differential equations in Banach spaces, Semigroups of operators: theory and applications (Newport Beach, CA, 1998), 352-363, Progr. Nonlinear Differential Equations Appl., 42, Birkhauser, Basel, 2000.
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Least Action for N-Body Problems with Quasihomogeneous Potentials Shih-liang Wen Department of Mathematics, Ohio University,Ohio 45701,USA
Shiqing Zhang Mathematical Department, Chongqing University, Chongqing 400044, China Abstract
Using variational minimization methods, we study the existence of a
noncollision or a generalized periodic solution for N-body problems with quasihomogeneous potentials, specially for N-body and N+l-body problems in R2k(k
^ 1), we study the geometric characterization for variational minimization solutions.
1. Introduction and Main Results N-body problems with quasihomogeneous potentials ([6], [12]-[14], [22]) are related with the motion of N point masses mi, •••, WN in RK (KS=1) under the action of the
potential -W(q) given by ±-
fo,-?)
Where U(q) = - jT ———'— li4*j±N q.-q. b v-i
(1.1)
(1.2)
tn,mt ^i
2
2
Where a,b>Q,a +b *0,a,/3>Q,a2 + p2 ^0,q = (ql,---,qN),qi e RK. The equations of the motion for the N-body problems with a potential -W(g) are given by m
,
dW(q) . „ *,i = l,---,N 8q,
(1.4)
Note that -W(q) is the classical Newtonian potential when a =0, P =1, or b =0, a=1 or a = P =1 ,and is a homogeneous potential when a =0 or 6 =0 or a = p. References [1H5], [8], [10H12], [15], [19]-[23] used variational methods to study the periodic solutions of N-body problems. The true motions of the celestial bodies should obey the Least Action Principle of Fermat-Maupertuis in some sense. Hence, we use variational minimizing methods to study the periodic solutions of N-body problems.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Serra-Terracini ([15]) used variational minimizing methods to study the existence of a noncollision periodic solution for 3-body problems with the classical Newtonian potential and a radial perturbation potential in R3. Long and Zhang ([10-12]) and Zhang-Zhou ([22]) and Chenciner and Desolneux ([3]) studied the shape of the solution minimizing the Lagrangian action integral on the T/2-antiperiodic or zero mean loop of class W1'2 (RJTZ, K ) for some N-body problems. Zhang and Zhou ([23]) studied the variational characterization of Lagrangian elliptical solutions with equilaterial triangle configurations for planar Newtonian 3-body problems, they proved that the regular minimizers of the Lagrangian action integral restricted on the periodic orbit space with three nonzero relative winding numbers are exactly the Lagrangian elliptical solutions. In this paper, we study the existence of a noncollision or a generalized periodic
solution for the systems (1 . 1 )-(! .4). For (1.1)-(1.4) in R2k (k^l), we also study the shape of the orbit minimizing its
Lagrangian action integral defined on the periodic orbit space with the same integral mean for each body during one period. Definition 1.1 ([2], [4]) Given T>0, Let q,, eWl-2([0,T],RK') ,we say q - (
of (1.1)—(1.4),if there hold: ( i ) S(q) = f e [0, T] J31
Theorem 1.1 For any given T>0,
I f ( i ) a > 2 , /?>2
or(ii) a>2, 0
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If one of the following three conditions holds: ( i ) a>2, p>2, (a, P) * (2, 2) ( i i ) a>2, 0 < p<2, (a, f t ) ^(2, 2)
(iii)
p>2, 0
Then there is a T>0 such that (1.1)-(1.4) has a noncollision T-periodic solution with energy ho If the following condition holds ( i v ) 0 < « < 2 , Q
Then there is a T>0 such that (1.1)-(1.4) has a generalized T-periodic solution with energy h . Theorem 1.3 For N-body problem (1.4) with a quasihomogeneous potential (1.1) in ), we define
=f If »i|9,|2* + {>(?)*.« e A
(1-6)
Then the global minimum point q(f)={q\(f), qi(t\ —, 1) whose configuration is the central configuration minimizing all configurations with potential f/and Fat any moment t and whose mass points rotate along circular orbits around the center ( the common integral mean) with the same constant angular velocity on a fixed plane. Theorem 1.4 Let ml =m2 =--- = mN >0, mN+l >0,
miqt
(t) = 0, qt (t) * qj (t), \
(1.7)
mtm. ^<w*qi-qj
Then the minimizer q=(qi(t),---,qN(t),qN+l(t)) T-periodic solution of
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
of I (q) on S is a classical
Where
W(q) = U(q) + V(q)
m m<
'
-Kitjt.N^q.-t
(1.10)
(1.12)
and satisfies that qt(i = l,---,N) moves on the circle centered at qN+l = 0 with a fixed angular velocity and a fixed motion plane and the configuration of ml,---,mN is the central configuration minimizing all configurations with potential U and V at any moment.
2. The Proof of Theorem 1.1 Given T>0, Let A = j? = (ql,-,qN)\ql,e Wl'\RITZ,RK) ;
Lemma 2.1 The functional fiq) in (2.2) attains its global minimum value on the closure A ofA, and the minimizer q(t) = (ql(t\---,qN(t}) is a generalized T-periodic solution for the system (!.!)-( 1.4)
Proof Since flq) is coercive and weakly lower-semi-continuous on A, so f(q) attauis the global minimum value on A, furthermore, similar to the proof of [1], [2], [4], [8], we know that the minimum point q(t) is a generalized solution for the system (1.1H1.4). Lemma 2.2 In the systems (1.1)-(1.4),
Then there holds the Gordon's strong force condition ([8]): there exists a function Ge C\RK - (0},R) and a neighborhood N of 0 in R* such that
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Lemma 23
([8] [2])Let {q"} be a sequence in A and q"—*qed\ ,then
Now we can prove Theorem 1.1:
Under the assumption ( i ) or ( ii ) or (iii) of Theorem 1 . 1 , by Lemma 2.2, Wy ( £ ) satisfies the Gordon's strong force condition, then Lemma 2.3 implies the generalized
solution q(t) obtained by Lemma 2.1 is a noncollision periodic solution, otherwise, then there is a minimizing sequence q" such that f(q") -> f(q) = -H» , this is contradiction
with inf \f(q\q e A} < +00 .
3. The Proof of Theoreml.2 We define , VxeM = \x = (je lf -, xN)e T, ^[-W(x)--W'(x)-x\dt
Where
e
j<. N,
'
(3.1)
=f
(3.2)
1], (Rk)N\ xt(t) *X](t [0, 1],
= 0}
(3.3)
Similar to the proof in [1] , we have Lemma 3.1 If x is a non-constant critical point for g(x) on M, then we can define TX):
T=
- \W\x)xdt
(3.4)
and q(t)=x(t/T) is a T-periodic solution of (1.1)-(1.5)0
Lemma 3.2 ([12])Fora>0
/3>0;a2 +fi2 * 0,he R, if g has a non-constant
critical point then we have (i) if (a, 13) € [2, + oo)2 \ {(2, 2)} then h>0, (ii) if («,/?) = (2, 2) thenh = 0 ,
(iii) if (a,/3) e [0, 2]2 \ {(2, 2), (0, 0)} then h<0. Lemma 33 For any integer k>0, N>2 and positive masses m\, •••WN, power index o>0
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
and q = (g,, ••-, qN) g (Rk)N the following homogeneous function with degree 0: / (3.5) attains global minimum on (Rfc)N, we denote this minimum by Da(N, m,C)a
Lemma 3.4 (Wirtinger's inequality) For any mj>0 and jc, e Wl-2([0,l],Rk) , (i=l,
N)if (3.6) then
(3.7) /=!
J=l
In the following , we'll prove Theorem 1 .2: (i)o>2, P>2, but a, p can't be 2 simultaneously By Lemma 3. 2, wehaveh>0» Hence 2 dt
By Lemma 3.4, We know that g(x) is coercive on M. =2, 0
(ii)0
Hence by Lemma 3.3, We have
By Wirtinger's inequality (Lemma 3.4) we have
i-—
Hence g(x) is coercive on M. Under the above two cases (i) and (ii), It's easy to know
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
that g(x)>0 and g(x) is weakly lower semi-continuous on M, and M is a weakly closed
subset. Hence g(x) attains the global minimum on M . The minimum value point is the required periodic solution. 4. The Proof of Theorem 1.3 In order to prove Theorem 1 .4, we also note that/^) is invariant under translations. Assume q = (ql,q2,-,qli)e A, Let l(t)
= ql(t)-[qlli = lA-,N
(4.1)
Then [x,]=0 and x=(x\, X2,'" ,XN) satisfies that Xi(t)ttj(t) (l
1
i
\x, | (4 2)
2;r 2
/i f V m \v -•r(^r) J,2,*/l*/ »
(4.3)
Let
(4.4)
Then
Where
(4.6)
We note that
hence fix) attaining its infimum if and only if the inequalities of Wirtinger and Coti Zelati's type and Jensen take equalities simultaneously. Hence we have (i) xt(t) = alcos~t + b,&Ja—t,ai,bteRu, a/2
(ii) U(x) \2mi \xi v/=i simultaneously,
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2
8 2
and
'
attains
their
minimum
Similar to the proof of [10, 11, 22], we deduce that the orbits Xi (t) (i=l, 2, —, N) locate the same fixed plane and must be circles whose centers are at the origin by (i) and (ii) and (iii). Hence qi(f)(i=\, 2,—, N) have the properties of Theorem 1.1 by (4.1).
5. The Proof of Theorem 1.4 N
mm
(N
Lemma 5.1 Let ^1=0, then % ' »+ ' >\ JTm, i=i |qt,|v U=i
+
22
•«„+,
(5.1)
and inequality (5.1) takes equality if and only if (5.2)
Now we prove Theorem 1.4: By the inequalities of Wirtinger and Lemma 3.3 and Lemma 5.1 we have
+ I —
-a/2
By Jensen's inequality we have Da(N,m,a)
Where note that ^(5) is strictly convex smooth function on S>0 and ^(5) ->• +00 as
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S ->• +00, hence ip(S) has a unique global minimizing Point S=So>0 and I(q) attains its
infimum if and only if all inequalities which we have used take equalities simultaneously, hence we have
(0
(ii) U(q)\J\mt\qin N
and
^'2
^i=l
attains their minimum simultaneously,
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solutions of the 3-body problem, Abstract, Chinese Science Bulletin 44(1999), 1653-1654. Acta Math Sinica, to appear.
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