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29Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 9, 2002
YJIK 517.977
OPTIMAL CONTROL PROBLEM OF SOME DIFFERENTIAL INCLUSION AND APPROXIMATIO N
A. Debinska-Nagorska, A. Just, Z. Stempien, K. Woznica
In this paper we present the optimal control problem governed by a variational inclusion with the monotone operator and a quadratic costfunctional. We apply standart Galerkin method to the approximation of the problem. After giving some results on the exis-tance of optimal control we shall prove the existance of weak condensation points of a set of solution of approximate problems. Each of these points is a solution of the initial optimization problem. Finally we shall give a simple example using the obtaned results.
§ 1. Introduction
The problems connected with inclusions were considered by many authors. The recent results were published by [2, 5, 9] and others. The recent results concerning optimal control for systems governed by the inclusions were published by [1, 7, 11].
We consider the optimal control problem governed by a second order differential inclusion with a linear continuous operator and a nonlinear multivalued maximal monotone operator. We apply standard Galerkin technique (see [4, 6]) and provide a convergence analysis.
The main result of our paper is the theorem proving the convergence of optimal values for approximated control problems to those of the original problem.
xPart of this work has been presented at the International Congress of Mathematicians held at Beijing in August 2002, thanks to a grant awerded from the Faculty of Technical Physics, Computer Sciences and Applied Mathematics of Technical University of Lodz.
© A. D§binska-Nagorska, A. Just, Z. Stempien, K. Woznica, 2002
Let V, H be two real Hilbert spaces such that V C H and the inclusion
mapping of V into H is continuous and compact. V* denotes the dual
space of V and H is identified with its own dual H* (see [2, 13]).
We shall consider the following nonlinear second order differential inclusion
y"{t) + Ay(t) + dip(y'{t)) 3 Bu(t), for t £ (0, T) (1)
with the initial conditions
2/(0) =y0 and y,(0)=y1 (2)
where y' denotes the generalised derivative on the interval (0,T) of the function y:[0,T]->V and 0 < T < oo ([8, 13]).
We assume that:
(i) The operator A : V —>• V* is a linear continuous symmetric and coercive operator i.e. there existes a positive constant a such that (Av,v) > a|M|y V v E V where (•,•) denotes the duality relation between the adequate spaces (see [2]).
(ii) dcp is the sub differential of a lower-semicontinuous proper convex function (p : V -^1U {oo} and 0 E d(p(0) (see [2, 13]).
(iii) The operator B : U —>• V* is a linear continuous operator and U is a Hilbert space.
(iv) The function / : [0, T] 3 t —>• f(t) = Bu(t) is of the class W1,2 (0, T; H) (see [2]), y0 e V, <p(2/i) < oo and {Ay0 + dip(yi)} n H / 0.
The differential inclusion (1) is equivalent to the next variational inequality (see [2])
{y"{t) + Ay(t) - Bu(t),z - y'{t)) > <p{y'{t)) - <p(z) , ,
a.e. t e (0, T) and V z £V
where (•,•) denotes the duality relation between the adequate spaces
[8].
Theorem 1. Let the assumptions (i) - (iv) be satisfied. Then there exists a unique solution y of the problem (1) - (2) such that: y E C([0,T];V), y' G (7([0, T]; H) D L°°(0, T;V), y" G L°°(0,T;H) (see [2]).
Let F denote the operator F : L2 (0, T; U) —>• L2 (0, T;V) x L2 (0, T; H) such that F{u) = {y,y'), where y is the solution of (1) - (2).
Lemma 1. If the assumptions (i) - (iv) are satisfied then the operator F is the Lipschitz map. Moreover, the operator F is weakly continuous map.
Proof. In the first part of the proof we shall present Lipschitz continuity of the operator F. Let F(u) = (y,yr) and F(u) = (y,y') for u,u E U. Using the monotonicity of the sub differential dip in (1) we obtain
(Bu(t) - y"(t) - Ay(t) - Bu(t) + y"(t) + Ay(t),y'(t) - y'(t)) > 0
and using linearity of the operators A and B we have
(y"(t) -y"(t),y>(t) -y'(i)) + (A(y(t) -y(t)),y(t) -y'(i)) <
< (B(u(t) -u(t)),y'(t) - y'(t)).
Hence
\jt + -W)),y'(t) ~y'(t)) <
< (B(u(t) - u(t),y'(t) -y'(t)).
By integration (4) over an arbitrary interval [0,t] C [0,T] with the assumptions (i) - (iv), and with 2ab < ja2 + eb2 for e > 0 and a, b E M (applying Schwartz’s inequality) we have
\\y\t)-y\m2H+*Mt)-m\\2v <
~ Cl (/ ~1I(s^uds + !0 (5)
for certain ci > 0 and a.a. t E [0,T]. From (5) by Gronwall’s inequality
(see [8]) we obtain
\\y'(t)-y'(t)\\2H + \\y(t)-y(t)\\2v <c2 f ||u(s)-u(s)||^ds (6)
Jo
for certain C2 > 0. Inequality (6) implies that the operator F is Lipschitz map. Thus we have proved the first part of the Lemma. Let a sequence (un) satisfy the following condition
un^u weaklyin L2(0,T;U) (7)
and yn satisfy differential inclusion (1) with u — un i.e
y"(t) + Ayn(t) + d<p(y'n(t)) 9 Bun(t) (8)
and initial conditions (2) i.e.
Vn{0)=yo, y'n(Q)=yi- (9)
From the Theorem 1 we know that the problem (8) - (9) has exactly one solution yn for n E N. From the assumption of Lemma and from the first part of the proof we obtain
\WnmH + \\ym2v<c\\un\\iHm) (io)
for certain c > 0 and a.a. t E (0, T). Further from (8) we infer that the subsequence (y”) is bounded in L2(0,T; H) too. From the assumption (7) and (10) follows that there exists a subsequence, which we also denote (yn)j converging weakly to an element y in L2{0, T; V) and respectively its subsequence (y'n) converges weakly to y1 in L2(0,T; H) and also subsequence (?/") converges weakly to y" in
L2(0, T; H). This implies that the operator F is weakly continuous map by the demiclosedness of dip and by the unique solution of the problem (1) - (2). □
§ 2. Optimal Control Problem
Let there be given a space of controls L2(0,T;i7) and elements ^e£2(0,T;tf).
The optimal control problem (P) can be stated as follows [3, 10]: find a control u° E L2(0,T; U) which minimizes the integral functional
J(y,u) = Ai/0T\\y(t) -yd\\2Hdt+
(11)
+a2 ¡0 \\y'(t) -y'dW2Hdt + /0T\\u(t)\\ldt,
where y = y(u) is a solution of (1) - (2) for u E L2 (0, T; U) and Ai, A2 > 0 and \\ + > 0.
We put 4>(u) = J(y(u),u). Using the definition of an optimal control u° we obtain that $(u°) = iniueL2(o,T-:u) $(u)-
Theorem 2. Let the assumptions (i) - (iv) be satisfied. Then the optimal control problem (P) has at least one optimal solution u° E L2 (0, T; U).
The proof of this Theorem is standard by applying a minimizing sequence for the functional J because the functional (11) is weakly lower-semicontinuous in L2(0,T;ii) x L2(0,T;i7).
§ 3. Approximation of the Control Problem
Let denote the approximate family of all finite-dimensional subspaces of the original space V i.e. W E implies W C V, dim W < oo and U~ V' family approximates the space H too. The approximation of space L2(0,T;V) is here understood as the family of spaces {L2(0,T; W)} (see [13]).
As an approximation of the control space U we assume a family of all finite-dimensional subspaces of the original space U i.e. Y C £/, dim Y < oo and \JreY = U'
We shall study the following approximated optimal problem (Pyvr): find a control UyW E L2(0,T;Y) which minimizes the cost functional
$(uy) = J(vw,uy) = Ai [ \\yw(t) ~ ydw\\2H^t + (12)
Jo
+A2 f \WwW — y'dw\\2H^ + [ \\uY(t)\\ijdt
Jo Jo
where yw = Vw(uy) is the solution of the inclusion
y'w(t) +Ayw(t) +d<p(y'w(t)) 3 BuY(t), fori E (0,T) (13)
with the initial conditions
Vw{ 0) = Vow and y,w( 0) = yiw (14)
where yow and yiw are the orthogonal projections of yo and yi onto W, ydw and y'dW are the orthogonal projections of yd and y'd onto L2(0, T; W) with the norm from the space L2{0, T;H).
Theorem 3. Under the assumption from Theorem 2 the optimal control problem (Pyw) has at least one optimal solution UyW.
The proof of this theorem can be made in the same way as the proof of Theorem 2 because the inclusion (13) with the initial conditions (14) has the unique solution yw = Vw{uy)>
From Lemma 1 we have the following corollary
Lemma 2. Let (uy) be a sequence of elements in L2(0,T;Y) and (yw) be a sequence of solutions of (13) - (14). If the assumptions of Lemma 1 are satisfied then the following conditions hold:
(a) If Uy —^ u weakly in L2( 0, T; [/) for dim W oo then yw y weakly in L2(0,T;F) and yw —>• y strongly in L2(0,T;ii) and y'w ^y' weakly in L2(0, T; H) for dim Y oo and dim W oo.
(b) If uy —>• u strongly in L2(0,T;i7) for dimF —>• oo then yw —>• y strongly in L2(0,T;F) and y'w y strongly in L2(0,T;ii) for dim Y —> oo and dim W —> oo.
The proof of parts (a) and (b) follows immediately from Lemma 1.
Let us now consider the problem of convergence of the approximation.
Theorem 4. Let the assumptions of (i) - (iv) be satisfied. Then there exist weak condensation points of a set of solutions of the optimal problems (Pyw) in L2(0, T; H) x L2(0, T; U) and each of these points is the solution of the optimal problem (P).
Proof. The sequence (UyW) 18 a minimizing sequence for functional (12). According to the approximation of the space U for u° (solution of problem (P)) there exists a sequence (uy) such that uy —>• u° strongly in L2(0,T;[/) for dimy —>• oo and (from Lemma 2) yw = yw(uy) y° = y(u°), y'w y0' strongly in L2(0,T; H) for dimF —>• oo and dim W oo where yw is a solution of the problem (13) - (14) for uy — uy. Since
= J{y°,U°) < J{Vw,UyW) < J(VW,UY)
where y^ = yw(uyW) 18 the solution of the problem (13) - (14) for uy = UyW' Then because the functional J is continuous on L2(0,T;ii) x L2(0, T; U) we have
lim J(vw,uyw) =
for dimy —> oo and dim W —> oo. The functional J is coercive, therefore the sequence (uyW) is bounded in L2(0,T; U). It follows that there exists a subsequence which we also denote (uyW) such that UyW u weakly in L2 (0, T; U) for dim Y oo and dim W —>• oo. Then Lemma 2 implies that y'w = yw(uyW) y'w(uYW^y' weakly in L2(0,T;H) for dimY ->• oo and dim W —> oo where y is a solution of the problem (1) - (2) for
u — u. The functional J is weakly lower-semicontinuous on L2(0,T; H) x L2(0,T;i7). Then we have
for dimy —> oo and dim W —> oo. This implies that u is one of the solutions of the optimal control problem (P). □
Example. We denote V = Hq(Q) and H = L2(0) where 0 Cn is an open bounded set with a sufficiently regular boundary T (see [8]) and Q = ft x (0,T).
We shall consider the following control problem:
The set C is any nonempty convex closed subset of i?g(ft), yo E i?o(ft) D ii2(ft) and yi E C. We assume that u E L2(Q). From Theorem 1 there exists a unique solution of the equation (15) y E C([0, T]; Hq (ft)) and || E L°°(0, T; iig (ft))n(7([0, T]; L2(ft)). Using the notations from Section 3 we transform the problem (15) to the system of differential inclusions
L2^f j,'U) in^ JiVw "> 'U'Yw) —
minimize J(y,u) = \\y(t)\\2L2{n)dt + \\u(t)\\2L2{n)dt
subject to
3 u(t, x) a.e. Q,
(15)
2/(0, a;) = Vo (x), = yi(x) a-e
y(t,x) = 0 a.e. T x (0,T)
= y1(x) a.e. 0,
where
[7]:
d2yw(t, x) _ -A d2yw(t,x) dt2 dx2 ^
(dyw{t,x)\
\—d^)3UY{t'x)
with the initial conditions
yw(fy = Vow and y'w( 0) = yiw
where yow and yiw are the orthogonal projections of yo(x) and yi(x) onto W.
The above system of differential inclusions is equivalent to the next systems described by functional differential inequalities
l^y^l _ En=i _ Uy(t)X))Zw(x) _
> X (9pwgt’x)) ~x(zw(x)) Vzw^W
with the initial conditions
Vw{ 0) = Vow and y'w( 0) = yiw-
Now, we can study the following optimal approximated problem: find a control UyW E Y which minimizes a cost functional
J{UW,UY) = \\yw\\2LV(Q) + \\UY\\2L2(Qy
The above example may be compared with the study of the vibrating string with an obstacle (see [12]).
References
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[11] Tiba D. Optimality Conditions for Distributed Control Problems with Nonlinear State Equation// SIAM J. Control Optim. 1985. V. 23(1), 85-110.
[12] Tiba D. Quelques remarques sur le contrôle de la corde vibrante avec obstacle// C. R. Acad. Sei. Paris. 1984. V. 229-1(13), 615-617.
[13] Zeidler E. (1990) Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators. New York: Springer—Verlag, 1990.
Institute of Mathematics, Technical University of Lodz,
PL-90-924 Lodz, Al. Politechniki 11, Poland
E-mail: debinska@ck-sg.p.lodz.pl
Université des Sciences et Technologies de Lille,
Polytech’Lille, Laboratoire de Mécanique de Lille
