数理解析研究所講究録
1187 巻 2001 年 27-36

27

OPTIMAL CONTROL PROBLEM FOR THE NONLINEAR
HYPERBOLIC SYSTEMSJONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

ABSTRACT. In this paper we study parameter optimal control monitored by nonlinear
hyperbolic systems. We show that for every value of the parameter, the optimal control
problem has a solution. Moreover we obtain the necessary optimality condition on thecontrol system.

1.

INTRODUCTION

The optimal control problems have been extensively studied by many authors [1.3,5,7.10,13
and reference there in] and also identification problem for damping parameters in the second order hyperbolic systems have been dealt with by many authors [6,8,12 and there
reference in]. In this paper, we consider the following control systems$y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=Bu+f\cdot(t, q)$

(1.1)

$\{$

$y(q, u)(0)=y_{\mathrm{U}}\in V,$ $y’(q, u)(0)=y_{1}\inH$and the cost functional given by the quadratic form
(1.2)

$J(q, u)= \frac{1}{2}||Cy(q, u)-z_{d}||_{M}^{2}$.

Here $A_{1}(t, q)$ , and $A_{2}(t, q)$ are differential operators containing unknown parameter $q\in Q$
and there are given by some bilinear forms on Hilbert spaces. $N^{*}g(Ny)$ is a nonlinear
term.

$B$

is a controller. $u\in U$ is a control,$f$

operator defined on an observation space $M,$

is a forcing term and $C$ is an observation$z_{d}$

is a desired value. The optimal con-trol problem subject to (1.1) nith (1.2) is to find an element

inf$J(q, u)=J(\overline{q},\overline{u})$$(q.u)\in Q\cross lI$$(\overline{q},\overline{u})\in Q\cross U$

such that. In this paper we will study the optimal control to the system(1.1) with (1.2) and the existence of weak solution for (1.1). It is not easy to find the
optimal control pairs$(\overline{q},\overline{\prime(x})$

belonging to a general admissible set$Q\cross U$of parameters andcontrols subject to (1.1) with (1.2). Hence we will show the existence of such$(\overline{q},\overline{u})$when1991 Mathematics Subject Classification.
.
Key words and phrases. Optimal control, nonlinear hyperbolic systems, quadratic cost function.
This work was supported by Brain Korea 21. 1999.
$93\mathrm{C}20,49\mathrm{J}20,49\mathrm{K}‘ \mathit{2}0$28
JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE$Q\cross U$

is a compact subset of a topological space. Recently, inspired by the optimal con-trol theoretical studies of Euler-Bernoulli Beam Equations with Kelvin-Voigt Damping,and Love-Kirchoff Plate Equations with various damping terms, the appeared numerous
paper studying optimal control theory and identification problems. In Banks et al. [4],Banks and Kunisch [5], they treated the existence of the optimal control (or minimizing
parameters) by using the methods of approximations, but they didn’t deal with the nec-

essary conditions (or characterizations) on them. When $A_{1}(t, q)\equiv\gamma A_{2}(t, q),$ $\gamma>0$ and$N^{*}g(Ny)=0$ in (1.1), the identification problem estimating

via output least-square$q$identification problem is studied by Ahmed $[1,2]$ based on the transposition methods. In
the nonlinear parabolic type case, Papageorgiou [$11_{\rfloor}^{-}$

treated with the optimal control

problems contained parameter and control. But we deal with the second order nonlinear
hyperbolic systems.

In specially. in this paper we study the optimal control (or minimizing parameters)
problems to (1.1) with (1.2) on the Gelfand five fold and the necessary conditions.

2.
Let $X$ be a real Hilbert spaces.

induced norm on X.
between$X^{*}$PRELIMINARIES$(\cdot, \cdot)_{X}$and$||\cdot||_{X}$the dual space of $X$ and$X^{*}$

denote the inner product and thedenotes the dual pairing,$\langle\cdot, \cdot\rangle_{XX}.$

and $X$ . Let us introduce underlying Hilbert spaces to describe the nonlinear

hyperbolic systems. Let $H$ be a real pivot Hilbert space, its normby$|\cdot|_{H}$spacesis denoted simply. Throughout this paper we assume there is a sequence of real separable Hilbert$V_{1},$$V_{2}^{*}arrow fV_{1}^{*}$

with$||\cdot||_{H}$$V_{2},$$V_{1}^{*},$$V_{2}^{*}$

forming a Gelfand quintuple satisfying. And also we assume that the embedding$||\phi||_{V_{2}}\leq c||\phi||_{V_{1}}$for$\phi\in V_{1}$and$V_{2}\mathrm{c}_{arrow}H$$V_{1}arrow\rangle$$V_{1}arrow\rangle$$V_{2}$$V_{2}arrow\rangle$$H\equiv H^{*}arrow\rangle$

is dense and continuous

is a densely compact embedding. From

now on, we write $V_{1}=V$ for convenient of notation. We assume that the equalities$\langle\phi, \varphi\rangle_{V^{*},V}=\langle\phi, \varphi\rangle_{V_{2}^{*},V_{2}}$for$\phi\in V_{2}^{*},$ $\varphi\in V$

and$\langle\phi, \varphi\rangle_{\mathcal{V}^{*},V}=(\phi, \varphi)_{H}$for $\phi\in H,$$\varphi\in V$.

We shall give an exact description of the nonlinear hyperbolic systems. We suppose that$Q$

is algebraically contained in a linear topological vector space with topology$Q_{\Gamma}.=(Q, \tau)$

space

$Y$

is compact. And also we suppose that

$U$

and

is compact subspace of Hilbert. Let $I=[0, T],$ $T\geq 0$ be fixed and $t\in[0, T]$ . Let

We will need the following hypotheses on the data.

$\tau$$q\in Q_{\tau}$.

29
OPTIMAL CONTROL PROBLEM FOR THE HYPEMBOLIC SYSTEMS.
$\mathrm{H}(\mathrm{A})$::$A_{i}$$I\cross Qarrow \mathcal{L}(V_{i}, V_{i})$is an operator $(i=1,2)$ .

(1) $a_{i}(t, q;\phi, \varphi)=a_{i}(t, q;\varphi, \phi)$ , where(2) There exists$c_{\vee i1}>0$such that(3) There exists$\alpha_{i}>0$

and(4) The function(5) There exists $c_{i2}>0$ such that

and$\mathrm{H}(\mathrm{f})$$\mathrm{H}(\mathrm{B})$$\mathrm{H}(\mathrm{N})$:such that$I\cross Qarrow V_{2}^{*}$

:
:

$N$

:

$V_{i}arrow H$$|a_{i}’(t, q;\phi, \varphi)|\leq c_{i2}||\phi||_{V_{i}}||\varphi||v_{i},$.
$\forall\phi\in V_{i}$.

$\forall\phi,$$\varphi\in V_{i},$where$’= \frac{d}{d\mathrm{f}}$.

is the forcing term such that $f(t, q)\in L^{2}(0, T;V_{2}^{*})$ .

is a linear operator such thatis constant and the range of $N$ on$g$.

is continuously differentiable in $[0, T]$ .

$a_{i}’(t, q:\phi, \varphi)=\langle A_{i}’(t, q)\phi, \varphi\rangle_{V_{i}}*,v_{i}$:$\varphi\in V_{i}$$a_{i}(t, q;\phi, \varphi)+\lambda_{i}|\phi|_{H}^{2}\geq\alpha_{i}||\phi||_{V_{i}}^{2},$: $B:Yarrow V_{2}^{*}$ is a bounded linear operator such that$k_{i}$$\mathrm{H}(\mathrm{g})$.

$f$$\forall\phi,$$|a_{i}(t, q;\phi, \varphi)|\leq c_{i1}||\phi||v_{i}||\varphi||v_{i},\forall\phi,$ $\varphi\in V_{i}$$\lambda_{i}\in R$$t\mapsto a_{i}(t, q;\phi, \varphi)$$a_{i}(t, q;\phi, \varphi)=\langle A_{i}(t, q)\phi, \varphi\rangle v_{i}*,v_{i},$$V_{i}$$B\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(Y, V_{2}^{*}))$$N\in \mathcal{L}(V_{\mathrm{i}}, H)$

with

.

$||N\varphi||\leq\sqrt{k_{i}}||\varphi||_{V_{2}}$,is dense in $H$ .

: $Harrow H$ is a continuous nonlinear mapping of real gradient$(\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l})\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}$such that(1) $||g(\varphi)||\leq c_{1}||\varphi||+(- 2,$(2)$\varphi\in H$$||g(\varphi)-g(\phi)||\leq C_{\backslash }‘’||\varphi-\phi||,$and for some constant$\varphi,$$\phi\in H$$c_{1},$$\mathrm{c}_{-2}$,and for some constant$C’.;$.

We consider the following problem for the nonlinear second order evolution equationsof the form :(2.1)$y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=f(t, q)$(2.2)$y(q)(0)=.\tau/\{)\in V,$ $y’(q)(0)=y_{1}\in H$

where$y’= \frac{d?/}{dt},$ $y”= \frac{d^{2}y}{dt^{2}}$,. We define a Hilbert space. which will be a space of solutions.

as$W(0, T)=\{y|y\in L^{2}(0, T;V), y’\in L^{2}(0, T:V_{2}), y’’\in L^{2}(0, T;V^{*})\}$

with an inner product$(y_{1},.y_{2})_{W((),T)}= \int_{0}^{T}\{(.\tau/\iota(t), y_{2}(t))_{\mathcal{V}}+(y_{1}’(t,),y_{\mathit{2}}’‘(t))_{V_{2}}+(y_{1}’’(t), y_{2}’’(t))_{V^{*}}\}dt$

and the induced norm$||y||_{W((),T)}=(||_{lj}’.||^{\frac{")}{L}}9((),\tau);V)+||_{l/’}’.||_{L^{9}((),T;1^{r_{9}})}^{2}.+||y^{\prime/}||_{L^{2}((\mathrm{I},T;V^{*})}^{2})^{\frac{1}{2}}$We denote by$D(\mathrm{O}, T)$

the space of distributions on $(0, T)$ .

.

30
JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

Definition 2.1. A function
and

$y$

(2.3)

$y$

is said to be a weak solution of $(2.1)-(2.2)$ if $y\in W(0, T)$satisfies$\langle y^{\prime/}(\cdot), \phi\rangle_{V^{*},V}+a_{2}(\cdot)q;y’(\cdot),$$\phi)+a_{1}(\cdot, q;y(\cdot),$ $\phi)+\langle g(N.y(\cdot)), N\phi\rangle_{H}=\langle].(\cdot, q), \phi\rangle_{V_{2}^{*},V_{2}}$for all $\phi\in V$ in the sense of(2.4)$D(\mathrm{O}, T)$,$y(q)(0)=y_{()}\in V,$ $\frac{dy}{db}(q)(0)=y_{1}\in H$ .

By Definition2.1 it is verified that a weak solution
(2.5)

$y$

of (2.1) satisfies$\int_{()}^{T}\langle y’’(t)+A_{2}(t, q)y’(t)+A_{1}(t, q)y(t)+N^{\lambda}g(Ny(t,)), \phi(t)\rangle_{\mathrm{I}/V_{2}}2^{\cdot}’ dt$$= \int_{()}^{T}\langle f(t, q), \phi(t,)\rangle_{V_{2}^{*},V_{2}}dt,$$\forall\phi)\in L^{2}(0, T;V_{2})$.

We state the existence and uniqueness results of a weak solution of $(2.1)-(2.2)$ .
Theorem 2.1. If $H(A),H(f),H(B),H(N)$ and $H(g)$ hold and$L(t)sat,isfyL(\cdot)\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(V_{2}, V_{2}^{*}))$

Then the equat,ion
(2.6)

$\{$

$y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=L(t)y+f(t, q)$$y(q)(0)=?/0\in V,$ $y’(q)(0)=y_{1}\in H$ ,has a unique we $aksolut,iony\in W(0, T)\cap C(\mathrm{O}, T;V)\cap C^{1}(0, T;H)$ .
weak solution for (2.6)$\dot{w}$$Her\cdot et,ht_{J}^{I}$

concep , of a$t$

defined as$\langle y’’(\cdot), \phi\rangle_{V^{*},V}+a_{2}(\cdot, q;y’(\cdot),$ $\phi)+a_{1}(\cdot, q;y(\cdot),$$=\langle L(\cdot)y(\cdot)+f(\cdot, q), \phi\rangle_{V_{2}V_{\underline{9}}}.,,$ $\forall\phi\in V$$wit,h$in $(0, T)$ ,in$\phi)+\langle g(Ny(\cdot)), N\phi\rangle_{H}$$t,h(^{\lrcorner}$

,

sens

$\rho_{J}$of $D’(0, T)$

the init, $ial$ conditions $y(q)(0)=y_{0}\in V,$ $y’(q)(0)=.l/1\in H$ .

PROOF. We can prove by using the method Lions [9] and Ha [8].
3. EXISTENCE OF BOTH PARAMETERS AND CONTROLS FOR OPTIMALITY
In this section we consider the optimal control problem for the following system:(3.1)$y^{\prime/}+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=Bu+f(t, q)$$\{$

in $(0, T)$$y(q, u)(0)=y_{()}\in V,$ $y’(q, u)(0)=y_{1}\in H,$ $q\in Q_{\tau},$ $u\in[I$.

Note that since there is a unique solution
well-defined mapping $y=y(q, u)$ of$y$$Q_{\tau}\cross[I$to (3.1) for given $(q, u)\in Q_{\tau}\cross[’$ , we have ainto $W(0, T)$ .

31
OPTIMAL CONTROL PROBLEM FOR THE HYPERBOLIC SYSTEMS.

We often call (3.1) the state equation and $y(q, u)$ the state with respect to (3.1). Let us
consider a quadratic cost functional attached to (2.6) as(3.2)$J(q, u)= \frac{1}{2}||Cy(q, u)-z_{d}||_{M}^{2},$ $(q, u)\in Q_{\tau}\cross U$

where $M$ is a Hilbert space of observations, $C\in \mathcal{L}(W(0, T),$ $M)$ is an observer anda desired value belonging to $M$ . Our main aim is to find(3.3)

satisfying$(\overline{q},\overline{u})\in Q_{\tau}\cross[\Gamma$$(\overline{q},\overline{u})$. We call$(\overline{q},\overline{u})$the optimal control to thesystem (3.1) and (3.2). Furthermore. we will give an assumption to$H(A)_{1}$is$J( \overline{q},\overline{u})=\min_{(q,u)\in Q_{\tau}\cross U}J(q, u)$and to give a characterization of suchand$z_{d}$$f\cdot$:$a_{i}(t, q:\phi, \varphi),$$i=1,2$:$qarrow a_{i}(t, q;\phi, \varphi)$

Note that for each:

$Q_{\tau}arrow R$$q\in Q_{\tau},$$\phi,$

is continuous for all$\varphi\in V_{i}$$f,$$\in[0, T],$ $\phi,$ $\varphi\in V_{i}$.

the following equalities hold :$|| \sup_{\varphi||_{V_{i}}=1}|a_{i}(t, q;\phi, \varphi)|=\sup_{||\varphi||_{V_{i}}=1}|\langle A_{i}(t, q)\phi, , \varphi\rangle_{\iota/V_{i}}*.|=||A_{i}(t, q)\phi||_{V_{i}^{*}}$

whence the assumption $H(A)_{1}$ and the above equality imply that,$||A_{\mathrm{i}}(t, q)\phi||_{V^{*}}$,

is contin-uous on .
$q$$H(f)_{1}$:$qarrow f\cdot(\cdot, q)$:$Q_{\tau}arrow V_{2}^{*}$

is continuous.

Lemma 3.1. If $H(A),H(f),H(B),H(N),H(A)_{1}$ and$L^{\infty}(0, T;\mathcal{L}(V_{2}, V_{2}^{*}))$$U,$$H(f\cdot)_{1}$hold and also $L(t)$ satisfy$L(\cdot)\in$. Then $y(q, u)$ is $str\cdot onglycont,inuo\uparrow\iota s$ on $(q, u),$ $i.e.,$ $y(q, u)\in C(Q_{\tau}\cross$$W(0, T))$ .

PROOF. It can be proved by using the method of$\mathrm{A}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{d}[2]$and$\mathrm{H}\mathrm{a}[8]$.

Theorem 3.1. If $H(A),H(f),H(B)_{\rangle}H(N),H(A)_{1}$ and $H(f)_{1}$ hold and also $L(t)sat,isfy$$L(\cdot)\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(V_{2}, V_{2}^{*}))$. Then there$?S$at least one optimal control$(\overline{q},\overline{u})$if$Q_{\tau}\cross$

[’

$is$

compact.

PROOF. It is clear from Lemma 3.1 and continuity of norm.

$\square$4. NECESSARY CONDITION OF OPTIMALITY FOR BOTH PARAMETERS AND CONTROLS
Here we present the necessary condition (the minimizing condition) for the optimal
controls$(\overline{q},\overline{u})\in Q_{\tau}\cross U$

to the system (3.1) with the cost functional $J(p, u)$ given by

32
JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE(3.2). If $J(p,u)$ is G\^ateaux differentiable atnecessary condition on
(4.1)

,

in the direction$(\overline{q},\overline{u})$.

Note that since $J(q, u)$ composed of the term$J(q, u)$, the$\forall(q, u)\in Q_{\tau}\cross U$

denotes the G\^ateaux derivative at$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})$$(q-\overline{q}, u-\overline{u})$$(q-\overline{q}, u-\overline{u})$is characterized by the following inequality$(\overline{q},\overline{u})$$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})\geq 0,$

where

in the direction$(\overline{q},\overline{u})$$y(q, \prime\prime x)$, the G\^ateaux differentiability offollows ffom that of $y(q, u)$ . Hence to obtain that of $y(q, u)$ we will need thefollowing condition:$H(A)_{2}$:$qarrow A_{i}(\cdot, q)$

is G\^ateaux differentiable for all$L^{2}(0, T;\mathcal{L}(V_{i}, V_{i}^{*}))$

at$H(g)_{1}$for all$q\in Q_{\tau}$$f_{J}$

and $DA_{i}(t, q)(p)\equiv DA_{i}(t, q;p)\in$, where $DA_{i}(t, q:p)$ denotes the G\^ateaux derivativein the direction of .
$p$

$q$

: For any
with$\varphi\in H$

the R\’echet derivative of$||g_{\varphi}(\varphi)||_{\mathcal{L}(H,H)}\leq c_{4}$, where$g_{\varphi}(\varphi)$$g$exists and satisfies$g_{\varphi}(\varphi)\in \mathcal{L}(H, H)$

is the R\’echet derivative of

at

$g$

$\varphi$and ( is$i_{4}$

constant.
$H(f)_{2}$: $qarrow f(t, q)$ is G\^ateaux differentiable for all and $f_{q}(t, q)p\equiv f_{q}(t, q;p)\in L^{2}(0, T, V_{2}^{*})$ ,$t$where $f_{q}(t, q;p)$ is G\^ateaux derivative at$q$in the direction of .
$p$

Lemma 4.1. Assume that the conditions in Theorem 2.1, $H(A)_{1},$ $H(A)_{2},$and $H(g)_{1}$ aoe satisfied.

direction$(q-\overline{q}, u-\overline{u}),$Then $y(q, u)$ is weakly G\^at,eaux differentiable$der\iota ot,et,heG\hat{a}tea^{l}\llcorner\iota x$derivative of $y(q, u)$ by$H(f\cdot)_{1},$$H(f\cdot)_{2}$$(q, u)$in the$af,$$z=Dy(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})$,which satisfies the following Cauchy problem:$z^{\prime J}+A_{2}(t,\overline{q})z’+A_{1}(t,\overline{q})z+N^{*}g_{y}(Ny(\overline{q},\overline{u}))Nz$$=-DA_{2}(t,\overline{q};q-\overline{q})y’(\overline{q},\overline{u})-DA_{1}(t,,\overline{q};q-\overline{q})y(\overline{q},\overline{u})$

(4.2)

$\{$

$+B(u-\overline{u})+f_{q}(t,\overline{q};q-\overline{q})$in $(0, T)$$z(0)=z’(0)=0$ .

PROOF. We can prove by using the method of Ahemd [2] and Park et al. [12].
By Lemma 4.1, the cost functional $J(q, u)$ is G\^ateaux differentiable at

direction(4.3)$(q-\overline{q}, u-\overline{u})$$(\overline{q},\overline{u})$$\square$in the, and so, the condition (4.1) is rewritten by$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})=\langle C^{*}\Lambda_{M}(C.y(\overline{q},\overline{u})-z_{d}).z\rangle_{W^{*}((),T),W((),T)}$$+\langle C^{*}\Lambda_{M}(Cy(\overline{q},\overline{?x})-z_{d}), y_{u}(\overline{q},\overline{u};u-\overline{u})\rangle_{W^{*}(\mathrm{U}’I’),W(\mathrm{U},T)},\geq 0,$$\forall(q)u)\in Q_{\tau}\cross[’$,

33
OPTIMAL CONTROL PROBLEM FOR THE HYPERBOLIC SYSTEMS.

where
of

$C$

(i)

$z$

is a unique weak solution to (4.2),

and$\Lambda_{M}$is the canonical isomorphism of $M$ onto$\langle\Lambda_{M}\phi, \phi\rangle_{M^{\kappa},M}=||\phi||_{M}^{2}$(ii)$C^{*}\in \mathcal{L}(M^{\lambda}, W^{*}(\mathrm{O}, T))$$||\Lambda_{M}\phi||_{M}\cdot=||\phi||_{M}$$M^{*}$

is the adjoint operator

in the sense that,for all $\phi\in M$ . In order to avoid the complexity of setting up

observation spaces, we consider the following two types of distributive and terminal value
observations in time sense. that is, the following cases :(i) we take(ii) we take$C_{1}\in \mathcal{L}(L^{2}(0, T;V_{2}),$$C_{2}^{\mathrm{v}}\in \mathcal{L}(H, M)$

4.1. The case where$M)$

and observer $z(q, \prime n)=C_{1}y(q, u)$ ;and observer $z(q, u)=C_{2}y(q, u)(T)$ .

$C_{1}\in \mathcal{L}(L^{\underline{y}}‘(0, T;V_{2}),$$M)$

In this case the cost functional is given by$J(q, u)= \frac{1}{2}||C_{1}y(q, u)-z_{d}||_{M}^{2},$ $\forall q\in Q_{\tau}\mathrm{x}U$,and then the necessary condition (4.3) is equivalent to(4.4)$\int_{\{)}^{T}\langle C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q}, ’\overline{\iota x})(t)-z_{d}), z(t)\rangle_{V_{2}^{*},V_{2}}dt$$+ \int_{\{\}}^{T}\langle C_{1}^{*}\Lambda_{M}(C\mathrm{i}/(\overline{q},\overline{u})(t)-z_{d}), .y|J(\overline{q},\overline{u};u-^{r}\overline{(x})\rangle_{1^{\gamma*}V_{2}},\rangle’ dt\geq 0,$

Let us introduce an adjoint state(4.5)$\eta(\overline{q}, ’\overline{lx})$,

satisfying$\eta^{\prime/}(\overline{q},\overline{\tau x})-A_{\underline{)}}‘(t,,\overline{q})\eta’(\overline{q},\overline{\tau x})+[(A_{1}(t,,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(Ny(\overline{q},\overline{\uparrow x})N)^{*}]r)(\overline{q}, ’\overline{(x})$$=C_{1}^{*}\Lambda_{M}(C’ 1^{(}’,/(\overline{q},\overline{\tau x})-z_{d})$$\mathit{7}\mathit{1}(\overline{q}, ’\overline{l/})(T)=0,$

Since$\forall(q, u)\in Q_{\tau}\cross U$$’/’(\overline{q},\overline{\tau x})(T)=0$,.

$C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q},\overline{\tau x})-z_{d})\in L^{\mathit{2}}‘(0, T;V_{2}^{*})$

and$A_{2}’(t,,\overline{q})\in L^{\propto)}(0, T;\mathcal{L}(V_{2}, V_{2}^{*}))$(4.5) is well-posed and permits a unique weak solution$\eta(\overline{q},\overline{u})\in W(0, T)$, the equation

if we consider

the change of the time variable as $farrow T-7$ . Multiplying (4.5) by , which is the weak$z$

solution to (4.2), integrating it by parts after integrating it on $[0, T]$ , we obtain$\int_{()}^{T}\langle\eta(\overline{q}, ’\overline{t/})(t), z^{\prime/}(t)+A_{\underline{)}}‘(t,\overline{q})z’(t)_{\mathrm{t}}[A_{1}(t,,\overline{q})+N^{*}g_{y}(Ny(\overline{q}, ’\overline{(x})(t))N]z(t,)\rangle_{VV}..dt$(4.6)$= \int_{()}^{T}\langle 77(\overline{q},\overline{?x})(f), -DA_{2}(t,\overline{q};q-\overline{q}).y’(\overline{q},\overline{u})(\dagger)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{V^{*}.V}dt$$+ \int_{(\}}^{T}\langle\eta(\overline{q}, ’\overline{lx})(t,), B(u-\overline{\tau x})(t)+f_{q}(t,,\overline{q};q-\overline{q})\rangle_{\mathfrak{l}\cdot,V}d\dagger\geq 0,$$\forall(q, u)\in Q_{\tau}\cross[’$.

34
JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

From (4.3) and (4.4), we obtain the inequality$\int_{0}^{T}\langle\eta(\overline{q},\overline{\prime(x})(t), z^{\prime/}(t)+A_{\underline{)}}‘(t,\overline{q})z’(t)+[A_{1}(t,\overline{q})+N^{\mathrm{t}:}g_{y}(N.y(\overline{q}, ’\overline{lx})(t))N_{\mathrm{J}^{\mathrm{I}}}^{\urcorner}z(t)\rangle_{1/1/}.*.\cdot dt$$+ \int_{()}^{T}\langle C_{1}y_{u}(\overline{q},\overline{u};q-\overline{q})(t,), C_{1}y(\overline{q},\overline{u})(t)-z_{d}\rangle d7$$= \int_{()}^{T}\langle\eta(\overline{q},\overline{\tau x})(t), -DA_{2}(t_{J},\overline{q};q-\overline{q})y’(\overline{q},\overline{u})(t)-[)A_{1}(t,\overline{q};q-\overline{q})_{l/}’.(\overline{q}, ?\overline{x})(t)\rangle_{V^{*}.V}dt$$+ \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{1V}.,dt$$+ \int_{\mathrm{U}}^{T}\langle C_{1}y_{I4}(\overline{q},\overline{u};q-\overline{q})(t_{J}), C_{1}y(\overline{q},\overline{u})(t)-z_{d}\rangle_{V^{*}.V}dt\geq 0,$[;.

$\forall(q, u)\in Q_{\tau}\cross$Here we used the inequality (4.4). Summarizing these we have the following theorem.

Theorem 4.1. Assume $t,hafH(A),$ $H(f),$$H(f)_{2},$$H(g)_{1}$$H(B),$

hold. Then the $opt,imalco7\iota t,r‘$)$H(N),$$l(\overline{q},\overline{\uparrow x})\dot{u}$$H(g),$$H(A)_{1},$$H(A)_{2},$charac $t,erized$ by staf, a$e$$H(f)_{1}$,$7tdadjoi_{7}\iota t$equations and inequality:$y”(\overline{q},\overline{u})+A_{2}(t,\overline{q})y’(\overline{q},\overline{u})+A_{1}(t,\overline{q})y(\overline{q},\overline{?x})+N^{*}g(N_{l/}’.(\overline{q},\overline{u}))=B\overline{?x}+f(t,\overline{q})$$\{$$y(\overline{q},\overline{u})(0)=y_{()}\in V,$ $y’(\overline{q},\overline{?x})(0)=y_{1}\in H$$i\tau\iota,$$(0, T)$,$\eta^{\prime/}(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+[(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(N.y(\overline{q}, ’\overline{u})N)^{*}]\eta(\overline{q},\overline{u})$$=C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q},\overline{u})-z_{d})$$\{$in $(0, T)$ ,$\eta(T,\overline{q})=0,$ $\eta’(T,\overline{q})=0$$\int_{\mathfrak{c})}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$$+ \int_{()}^{T}\langle C_{1}y_{u}(\overline{q},\overline{\tau x};u-\overline{?x})(t), C_{1}.y(\overline{q},\overline{u})(t)-z_{d}\rangle_{1^{r_{*}},V}/dt$$\{$$\geq\int_{(\}}^{T}\langle\eta(\overline{q},\overline{\prime u})(t), DA_{2}(\mathrm{r}_{J},\overline{q};q-\overline{q})y’(\overline{q}, \tau\overline{x})(t,)+DA_{1}(t,\overline{q};q-\overline{q})’.l/(\overline{q},\overline{7\mathrm{J}})(t)\rangle_{V^{*},V}dt$,.

$\forall(q, u)\in Q_{\tau}\cross U$$\square$4.2. The case where$C_{2}\in \mathcal{L}(H, M)$

In this case the cost functional is given by$J(q, u)= \frac{1}{2}||C_{2}y(q, u)(T)-z_{d}||_{M}^{2},$ $(q. n)\in Q_{\tau}\cross U$

and then the necessary condition (4.3) is equivalent to(4.7)$(C_{2}^{*}\Lambda_{M}(C_{2}y(q, u)(T)-z_{d}),$ $z(T))_{H}$$+(C_{2}^{*}\Lambda_{M}(C_{2}y(q, u)(T)-z_{d}),$ $y_{u}(\overline{q},\overline{u}:u-’\overline{(x})(T))_{H}\geq 0,$$\forall(q, u)\in Q_{\tau}\cross U$.

35
OPTIMAL CONTROL PROBLEM FOR THE HYPERBOLIC SYSTEMS.

Let us introduce an adjoint state$\eta(\overline{q},\overline{u})\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\infty \mathrm{n}\mathrm{g}$$\eta^{\prime/}(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+\lfloor\lceil(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))$$+(N^{*}g_{y}(Ny(\overline{q},\overline{u})N)^{*}]\eta(\overline{q},\overline{\tau x})=0$

(4.8)

$\{$

$\prime\prime)(\overline{q},\overline{u})(T)=0$,$\eta’(\overline{q},\overline{u})(T)=-C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})$.

It follows by the same reason as the case 4.1 that there is a unique weak solution$\eta(\overline{q},\overline{u})\in$$W(0, T)$ . because $C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})\in H$ .

Theorem 4.2. We assume $t,hatH(A),$ $H(f),$ $H(B,),$ $H(N),$ $H(g),$ $H(A)_{1},$ $H(A)_{2},$$H(f)_{2}$and $H(g)_{1}$ hold.

Then $t,heopt,imal$ control$(\overline{q},\overline{u})$$H(f)_{1_{\rangle}}$is characterized by state andadjoint, $equat,ions$ and $inequalit,y$ :$y”(\overline{q},\overline{u})+A_{2}(t,\overline{q})y’(\overline{q},\overline{u})+A_{1}(t,\overline{q})y(\overline{q},\overline{u})+N^{*}g(Ny(\overline{q},\overline{u}))=B\overline{u}+f\cdot(t, q)$$\{$$y(\overline{q},\overline{u})=y_{()}\in V,$$y’(\overline{q},\overline{u})=y_{1}\in H$

in $(0, T)$,$\eta’’(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+[(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(Ny(\overline{q},\overline{u})N)^{*}]\eta(\overline{q},\overline{u})(T)=0$$\{$$\eta(\overline{q},\overline{u})(T)=0$,,$\eta’(\overline{q},\overline{\tau x})(T)=-C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{\prime lx})(T)-z_{d})$$(C_{2}^{*}(C_{2}y( \overline{q},\overline{u})(T)-z_{d})_{H}+\int_{\{)}^{T}\langle B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q}), \eta(\overline{q},\overline{u})(t)\rangle_{V^{*},V}dt$$\{$$\geq\int_{()}^{T}\langle DA_{2}(t,\overline{q};q-\overline{q})y’(\overline{q},\overline{u})(t)+DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t), \eta(\overline{q},\overline{u})(t)\rangle_{V^{*},V}dt,$$\forall q\in Q_{\tau}$.

PROOF. We prove the inequality condition of optimal control only. Multiplying (4.8)
by , which is a weak solution to (4.2), integrating it by parts after integrating it on $[0, t]$ .
$z$

we obtain$\int_{()}^{T}\langle\eta(\overline{q},\overline{u})(t), z^{\prime/}(t)+A_{2}(t,\overline{u})z’(t)+[(A_{1}(t,\overline{q})+N^{*}g_{y}(Ny(\overline{q},\overline{u})(t)N]z(t)\rangle_{V^{*},V}dt$$+(z(T), \eta’(\overline{q},\overline{\tau x})(T))_{H}$$= \int_{()}^{T}\langle r/(\overline{q},\overline{?x})(t), -DA_{2}(t,\overline{q};q-\overline{q}).y’(\overline{q},\overline{u})(t)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{VV}.,dt$$+ \int_{()}^{T}\langle\eta(\overline{q},\overline{u})(t_{J}))B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$$+(z(T), -C_{\underline{9}}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d}))_{H}=0$.

36
JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

Hence from (4.7) and (4.8) we conclude that$(z(T), C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d}))_{H}+(y_{u}(\overline{q},\overline{u};u-\overline{u})(T),$ $C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})_{H}$$= \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), -DA_{2}(t,\overline{q})q-\overline{q})y’(\overline{q},\overline{u})(t)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{VV}.,dt$$+ \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$$+(y_{\alpha}(\overline{q},\overline{u};u-\overline{u})(T),$$C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})_{H}\geq 0,$$(q, u)\in Q_{\tau}\cross U$.
$\square$

REFERENCES[1] N.U. Ahemd, Necessary conditions of optimality for a clas ‘9 of second order hyperbolic systems withspatially dependent controls in the coefficients, J. Optim. Theory and Appl., (1982), Vol. 38, pp.

423-446.

[2] N.U. Ahemd, Optimization and identification of systems govemed by $evol?4tion$ equabions on Banach
space, Pitman research notes in Mathematics series, Longrrlan Scientific&Technical 184, 1988.
[3] H.T. Banks, D.S. Gilliam and V.I. Shubov, Grobal solvability for damped abstmct nonlinear hyperbolic
systems, Differential and integral equations, (1997), Vol. lt). No. 2, pp. 307-332.
[4] H.T. Banks, K. Ito and Y. Wang, Well posedness for damped second order systems with unbounded
input operators, Center for research in scientific computation, North Carolina state univ., 1993.
,[5] H.T. Banks and K. Kunisch, Estimations Techniques for $d?str\cdot ibuted$ parameter systems.
$\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\ddot{a}\dagger \mathrm{l}\mathrm{s}\mathrm{e}\mathrm{r}$1989.

and Tcchnol[6] R. Dautray and J.L. Lions, Mathemabicd Analysis and Numerical Methods forogy, Springer-Verlag, 5, Evolution Problerns, 1992.
[7] J.H. Ha and S.I. Nakagiri, Eristence and Regularity ofsolutions for sernil.inear second 07 derevolution equations, Funkcialaj Ekvacioj, (1998), Vol. 41, pp. 1-24.
[8] J.H. Ha, Optimal control problems for hyperbolic distributecl parameter systems $e?thdamp?7\iota g$ terms.
Doctoral Thesis, Kobe Univ., Japan, 1996.
[9] J.L. Lions, Optimal control of systems governed by partial Differential $Equat_{?onS}$ , Springer-Verlag,
New York, 1971.
[10] O.A. Ladyzhenskaya, The boundary value problems of Mathematical physics,-Verlag, New
York, 1984.
. optirnality in on[11] A.Papageorigiou and N.S. Papageoriou, Neoessary and sufficientlinear distributed parameter systems $w?th^{J}ua\dot{n}able$ initial stat , J. Math. Soc. Japan. (1990), Vol. 42,
No. 3, pp. 387-396[12] J.Y. Park, J.H. Ha and H.K. Han, Identification problem for damping pararnetars
darnp $‘d$second order systems. J. Korea Math. Soc., (1997), Vol. 34, No. 4, pp.1-14.
[13] R. Temam, Infinite-Dimens?onal $Dynam?cal$ systems in Mechanic.9 and $phys?cs,$-Verlag, New
York, 1988.
$\tau scien\mathrm{r}e$$w\epsilon^{}ak$$\mathrm{s}_{\mathrm{P}^{\mathrm{r}\mathrm{i}\mathrm{r}1}\mathrm{g}\mathrm{e}\mathrm{r}}$$condit^{t}ionsf\dot{\mathrm{o}}r$$\mathcal{T}1$$‘$$i_{7}\iota linea7^{\cdot}$$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{r}\iota \mathrm{g}\mathrm{e}\mathrm{r}$DEPARTMENT OF MATHEMATICS, PIJSAN NATI
-mail address: jyepark@hyowon.pusan.ac.kr$E$$()\mathrm{N}\mathrm{A}\mathrm{L}\mathrm{I}\overline{\mathrm{I}}\mathrm{N}\mathrm{I}\mathrm{V}\mathrm{E}*\mathrm{I}\mathrm{T}\mathrm{Y}$, PUSAN$609- 73^{\ulcorner}.$).

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