数理解析研究所講究録
1197 巻 2001 年 137-144

137

OPTIMAL CONTROL FOR SEMILINEAR ABSTRACTEQUATIONS OF PARABOLIC TYPE

大阪大学大学院工学研究科柳相旭 (Sang-Uk Ryu), 八木厚志 (Atsushi Yagi)

Graduate School of Engineering,Osaka Univ.

1. INTRODUCTION
In the preceding paper [8], the authors studied the optimal control problems for
the Keller-Segel equations. In that paper we showed the existence of optimal control
and the first order necessary condition by formulating the Keller-Segel equations as asemilinear abstract equation. Many papers have already been published to study thecontrol problems for nonlinear parabolic equations. In the books Ahmed [1] and Barbu[2], some general frameworks are given for handling the semilinear parabolic equations
with monotone perturbations. In [1] the nonlinear terms are monotone functions withlinear growth, and in [2] they are generalized to the multivalued maximal monotone
operators determined by lower semicontinuous convex functions. Papageorgiou [7] andCasas et al. [3] have studied some quasilinear parabolic equations of monotone type.
This note is the generalization of [8] as a semilinear abstract equation of non-monotone
type.
,
Notations. denotes the sets of real numbers. Let be an interval in R.
$1\leq p\leq\infty$ , denotes thespace of measurable functions in with values in a Hilber
space .
denotes the space of continuous functions in with values in . Let$D(I)$ denote the space of-functions with compact support on and $D’(I)$ denotethe space of distributions on . For simplicity, we shall use a universal constant $C$ todenote various constants which are determined in each occurrence in a specific way by$M$ , and so forth. In a case when $C$ depends also on some parameter, say, it will be.
denoted by$I$$\mathbb{R}$$L^{p}(I;\mathcal{H})$$I$$L^{p}$$\mathcal{H}$$I$$C(I;\mathcal{H})$$\mathcal{H}$$I$$C^{\infty}$$I$$\theta$$\delta,$$C_{\theta}$

2. THE FORMULATION OF PROBLEM
and be two separable real Hilbert spaces with dense and compact embedding
and denoting the dual space of by , we
. Identifying
and its dual.
and the norm byby. We denote the scalar product ofhave
The duality product between $v’$ and which coincides with the scalar product of on, respectively.
andby, and the norms of andis denoted by.
ofsubsetclosed,boundedandconvexandis
We consider the following Cauchy problem

Let$\mathcal{V}(-\rangle$$\mathcal{V}$$\mathcal{H}$$\mathcal{H}$$\mathcal{H}’$$\mathcal{H}$$\mathcal{V}$$\mathcal{V}arrow \mathcal{H}arrow \mathcal{V}’$$\mathcal{H}$$|\cdot|$$(\cdot, \cdot)$$\mathcal{H}$$\mathcal{V}$$\mathcal{H}\cross \mathcal{H}$$\mathcal{U}=L^{2}(0, \tau;\mathcal{V}’)$(E)$\langle\cdot, \cdot\rangle$$\mathcal{V}$$\mathcal{V}’$$||\cdot||$$||\cdot||_{*}$$\mathcal{U}$$\mathcal{U}_{ad}$$\frac{dY}{dt}+AY=F(Y)+U(t)$ ,$\{$$Y(0)=Y_{0}$$\mathcal{V}’$$0<t\leq T$,

138
in the space . Here, is thedefinite self-adjointoperato-r of defined by asymmetric sesquilinear form $a(Y, Y)$ on, which satisfies$A$$\mathcal{V}’$$\mathcal{H}$$\mathrm{p}\underline{\mathrm{o}\mathrm{S}}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}$$\langle AY,\overline{Y}\rangle=a(Y, Y)$$\mathcal{V},$(a.i)$|a(Y,\overline{Y})|\leq M||Y||||\overline{Y}||$(a.ii)$a(Y, Y)\geq\delta||Y||^{2}$

with some and $M>0.$continuous function from$\delta$,,$Y,\overline{Y}\in \mathcal{V}$,$Y\in \mathcal{V}$is also a bounded operator fromtosatisfying$A$$\mathcal{V}$$\mathcal{V}$to$\mathcal{V}’$.

$F(\cdot)$is a given:$[0, \infty)arrow$$\mathcal{V}’$(f.i) For each $\eta>0$ , there exists an increasing continuous function
such that$||F(Y)||_{*}\leq\eta||Y||+\phi\eta(|Y|)$ ,$\phi_{\eta}$$[0, \infty)$$Y\in \mathcal{V},\cdot$(f.ii) For each $\eta>0$ , there exists an increasing continuous function
such that: $[0, \infty)arrow$$\psi_{\eta}$$[0, \infty)$$||F(\overline{Y})-F(Y)||_{*}\leq\eta||\overline{Y}-Y||+(||\overline{Y}||+||Y||+1)\psi_{\eta}(|\overline{Y}|+|Y|)|\overline{Y}-Y|$$U(\cdot)\in L^{2}(0, \tau;\mathcal{V}’)$is a given function and$Y_{0}\in \mathcal{H}$,$\overline{Y},$$Y\in \mathcal{V}$.

is an initial value.

We then obtain the following result (For the proof, see Ryu and Yagi [8]).

Theorem 2.1. Let (a.i), (a.ii), (f.i), and (f.ii) be satisfied.
and, there exists a unique weak solution$L^{2}(0, \tau;\mathcal{V}’)$

Then, for any $U\in$$Y_{0}\in \mathcal{H}$$Y\in H^{1}(0, T(Y0, U);^{v’)}\cap C([0, \tau(Y0, U)])\mathcal{H})\cap L^{2}(0, T(Y_{0}, U);\mathcal{V})$to (E), the number $T(Y0, U)>0$ is determined by the norms$||U||_{L(;}20,\tau v’$)and$|Y_{0}|$.

In this section we are concerned with the following problem(P)

Minimize $J(U)$ ,where the cost functional $J(U)$ is of the form
$J(U)= \int_{0}^{S}||DY(U)-Y_{d}||2dt+\gamma\int_{0}^{S}||U||_{*}^{2}dt$

Here, $Y(U),$interval $[0, S]$ .

,$U\in u_{ad}$.

, is the weak solution of (E) and is assumed to exist on a fixed
is a bounded operator frominto
andis a fixed element of
is a nonnegative constant.

$U\in \mathcal{U}_{ad}$$L^{2}(0, S, \mathcal{V})$.

$\gamma$$D$$\mathcal{V}$Remark. Let
be fixed. By Theorem 2.1, for$[0, T(U)]$ with $T(U)>0$ depending on$Y_{0}\in \mathcal{H}$$\mathcal{V}$$U\in \mathcal{U}_{ad},$$||U||L^{2}(0,T;\mathcal{V}’)$$\mathcal{U}_{ad}\}$.

We prove the following theorem.

$Y_{d}$$Y(U)$ exists

on the interval. Hence, $0<S \leq\inf\{T(U);U\in$139
Theorem 2.2. There exists an optimal control$\overline{U}\in \mathcal{U}_{ad}$$J( \overline{U})=\min_{\in U\mathcal{U}_{a}d}J(U)$for (P) such that.

Proof. The proof can be carried out in the same way as that of Theorem 2.1 (see [8,Theorem 2.1]). As it is standard (cf. [2, Chap. 5, Proposition 1.1] and [6, Chap. III,
Theorem 15.1]), we will only sketch.
Letbe a minimizing sequence such that
. Since$\{U_{n}\}\subset \mathcal{U}_{ad}$$\lim_{narrow\infty}J(U_{n})=\min_{U\in \mathcal{U}_{ad}}J(U)$is bounded, we can assume that
. For simplicity,
weakly in$Y(U_{n})$we will write
instead of the solution,of (E) corresponding to$U_{n}arrow\overline{U}$$\{U_{n}\}$$L^{2}(0, S;\mathcal{V}’)$$Y_{n}$$U_{n}$$\frac{dY_{n}}{dt}+AY_{n}=F(Y_{n})+U_{n}(t)$$0<t\leq S$ ,

,

$\{$

$Y_{n}(0)=Y_{0}$.

Taking the scaler product of the equation and$Y_{n}$, we obtain that.

$\frac{1}{2}\frac{d}{dt}|Y_{n}(t)|^{2}+\langle AY_{n}(t), Y_{n}(t)\rangle=\langle F(Y_{n}(t), Y(nt)\rangle+\langle U_{n}(t), Y_{n}(t)\rangle$

Then, from (a.ii) and (f.i),$\frac{1}{2}\frac{d}{dt}|Y_{n}(t)|^{2}+\delta||Y_{n}(t)||^{2}\leq\eta||Y_{n}(t)||^{2}+\{\phi_{\eta}(|Y_{n}(t)|)+||U_{n}(t)||*\}||Y_{n}(t)||$.

With some increasing, locally Lipschitz continuous function $\phi:[0, \infty)arrow[0, \infty)$ , it follows that$\frac{d}{dt}|Y_{n}(t)|^{2}+\delta||Y_{n}(t)||^{2}\leq\phi(|Y_{n}(t)|^{2})+\frac{8}{\delta}||U_{n}(t)||^{2}*$(2.1)$|Y_{n}(\mathrm{o})|^{2}=|Y_{0}|^{2}$

Let
that

’

$0<t\leq S$ ,

$\{$

.

$z_{n}(t)=|Y_{n}(t)|^{2}- \frac{8}{\delta}\int_{0^{t}}||U_{n}(s)||_{*}^{2}ds,$$0\leq t\leq S$. Since$\frac{dz_{n}}{dt}\leq\phi(Z_{n}+8C\delta-1)$$\int_{0}^{S}||U_{n}(s)||_{*}^{2}ds\leq C$, it follows.

On the other hand, let $z(t)$ be a solution to the ordinary differential equation$\frac{dz}{dt}=\phi(_{Z}+8C\delta-1)$,$0\leq t\leq S$

,

$\{$

$z(0)=|Y_{0}|^{2}$.

Then, by the theorem of comparison, $z_{n}(t)\leq z(t)$ for all $0\leq t\leq S$ . Hence,$||z||_{c(}[0,S])+8C\delta-1$ .
. As a consequence, it follows
is thus bounded in
The sequencealso. Moreover, from (f.i), $\{dY_{n}/dt\}$ isis bounded infrom (2.1) that$|Y_{n}(t)|^{2}\leq$$\{Y_{n}\}$$\{Y_{n}\}$$L^{\infty}(\mathrm{O}, s;\mathcal{H})$$L^{2}(0, s;\mathcal{V})$140bounded inthat$L^{2}(0, S;\mathcal{V}^{;})$. Therefore, choosing a subsequence if necessary, we can assume$Y_{n}arrow\overline{Y}$$\frac{dY_{n}}{dt}arrow\frac{d\overline{Y}}{dt}$Since$\mathcal{V}$weakly in
weakly in,$L^{2}(0, s;\mathcal{V})$$L^{2}(0, S;\mathcal{V}’)$.

is compactly embedded in 7#, it is shown by [5, Chap. 1, Theorem 5.1] that(2.2)

$Y_{n}arrow\overline{Y}$strongly in$L^{2}(0, S, \mathcal{H})$.

Let us veriP thatis a solution to (E) with the control, and put $\Phi(t)=\xi(t)V$ . Then,$\overline{Y}$$\overline{U}$. Let$\xi\in D(\mathrm{O}, S)$

and$V\in \mathcal{V}$$\int_{0}^{S}\langle Y’(t), \Phi(t)n\rangle dt+\int_{0}^{S}\langle AY_{n}(t), \Phi(t)\rangle dt$$= \int_{0}^{S}\langle F(Y_{n}(t), \Phi(t)\rangle dt+\int_{0}^{S}\langle U_{n}(t), \Phi(t)\rangle dt$

Let here$n$.

tend to infinity. It is then observed from (f.ii) that$\int_{0}^{S}|\langle F(Y_{n}(t)-F(\overline{Y}(t), \Phi(t)\rangle|dt\leq\eta\int_{0}^{S}||Y_{n}(t)-\overline{Y}(t)||||\Phi(t)||dt$$+ \int_{0}^{S}(||Y_{n}(t)||+||\overline{Y}(t)||+1)\psi_{\eta}(|Y_{n}(t)|+|\overline{Y}(t)|)|Yn(t)-\overline{Y}(t)|||\Phi(t)||dt$

where $\eta>0$ is arbitrary. From (2.2) it is seen thatas $narrow\infty$ . Therefore, we obtain that$\int_{0}s\langle F(Yn), \Phi(t)\rangle dt$,

converges to$\int_{0}^{S}\langle F(\overline{Y}(t)), \Phi(t)\rangle dt$$\int_{0}\langle\overline{Y}’(t), \Phi(t)s\rangle dt+\int_{0}^{S}\langle A\overline{Y}(t), \Phi(t)\rangle dt$$= \int_{0}^{S}\langle F(\overline{Y}(t), \Phi(t)\rangle dt+\int_{0}^{S}\langle\overline{U}(t), \Phi(t)\rangle dt$.

This then shows thatsatisfies the equation of (E) for almost all $t\in(0, S)$ . In a
, note from [4, Chap. XVIII, Theorem 1]
similar way it is also shown that
. Hence,thatis the unique solution to (E) with the control , that.
is,
, we have:inSince $Y_{n}-Y_{d}$ is weakly convergent to$\overline{Y}(t)$$\overline{Y}(0)=Y_{0}$$\overline{Y}$$\overline{Y}\in C([0, S];\mathcal{H})$$\overline{U}$$\overline{Y}=Y(\overline{U})$$\overline{Y}-Y_{d}$$L^{2}(0, s;\mathcal{V})$$\min_{U\in \mathcal{U}_{a}d}J(U)\leq J(\overline{U})\leq\varliminf J(U_{n})=\min_{\in U\mathcal{U}_{a}d}J(U)$$narrow\infty$Hence,$J( \overline{U})=\min_{U\in \mathcal{U}_{a}d}J(U)$.

$\square$.

141
3. FIRST ORDER NECESSARY CONDITIONIn this section, we show the first order necessary condition for the Problem (P). We, respectively. In order toandbydenote the scalar products in andthe necessary conditions of optimality, we need some additional assumptions:$\mathcal{V}’$$\mathcal{V}$$\langle\cdot, \cdot\rangle_{\mathcal{V}’}$$\langle\cdot, \cdot\rangle_{\mathcal{V}}$is Fr\’echet differentiable and for each $\eta>0$ , there:(f.iii) The mapping
exists an increasing continuous functions , l ノ : $[0, \infty)arrow[0, \infty)$ such that
$F(\cdot)$$\mathcal{V}arrow \mathcal{V}’$$\mu_{\eta}$$|\langle F’(Y)z, P\rangle|\leq\{$$\eta||Z||||P||+(||Y||+1)\mu\eta(|Y|)|z|||P||$ ,$Y,$ $Z,$ $P\in \mathcal{V}$,$Y,$ $Z,$ $P\in \mathcal{V}$$\eta||Z||||P||+(||Y||+1)\mu_{\eta}(|Y|)||Z|||P|$$I^{\text{ノ}}(|Y|)||Z||||P||$(f.iv)$F’(\cdot)$is continuous from$\mathcal{H}$

into

,

$Y,$ $Z,$ $P\in \mathcal{V}$$\mathcal{L}(\mathcal{V}, \mathcal{V}’)$

,
,

.

.

Proposition 3.1. Let (a.i), (a.ii), (f.i), (f.ii), (f.iii), and (f.iv) be satisfied. The mapping$Y$ :is G\^ateaux differentiable with respect$Y’(U)V=Z$ is the unique solution into U. Forof the problem$\mathcal{U}_{ad}arrow H^{1}(0, s;\mathcal{V}’)\cap C([0, S]).\mathcal{H})\cap L^{2}(0, S, \mathcal{V})$$H^{1}(0, S;\mathcal{V}’)\cap C([0, S];\mathcal{H})\cap$$V\in \mathcal{U}_{ad},$$L^{2}(0, S, \mathcal{V})$$\frac{dZ}{dt}+AZ-F’(Y)Z=V(t)$ ,(3.1)$0<t\leq S$ ,$\{$$Z(0)=0$ .

Proof. Let$U,$

and $0\leq h\leq 1$ . Let, respectively.

$V\in \mathcal{U}_{ad}$ing to $U+hV$ and$U$$Y_{h}$

Step 1. $Y_{h}arrow Y$ strongly in $C([0, S];\mathcal{H})$ as
satisfies

and $Y$ be the solutions of (E) correspond$harrow \mathrm{O}$. Let $W=Y_{h}-Y$ . Obviously, $W$$\frac{dW}{dt}+AW-(F(Y_{h}(t))-F(Y(t)))=hV(t)$ ,(3.2)$0<t\leq S$ ,$\{$$W(0)=0$ .

Taking the scalar product of the equation (3.2) with $W$ , we obtain that$\frac{1}{2}\frac{d}{dt}|W(t)|^{2}+\langle AW(t), W(t)\rangle=\langle F(Y_{h}(t))-F(Y(t\mathrm{I}), W(t)\rangle+\langle hV(t), W(t)\rangle$.

Using (a.ii) and (f.ii), we have$\frac{1}{2}\frac{d}{dt}|W(t)|^{2}+\delta||W(t)||2$$<\underline{\delta}||W(t)||^{2}+(||Y_{h}(t)||^{2}+||Y(t)||21+)\psi_{\frac{\delta}{4}()W}|Y_{h}(t)|+|Y(t)|2|(t)|^{2}$$-2$$+4h^{2}\delta^{-1}||V(t)||_{*}^{2}$.

142
Therefore,(3.3)$\frac{1}{2}|W(t)|^{2}+\frac{\delta}{2}\int_{0}^{t}||W(s)||2ds$$\leq\int_{0}^{t}(||Y_{h(S})||^{2}+||Y(S)||^{2}+1)\psi\frac{\delta}{4}(|Y_{h}(_{S})|+|Y(s)|)2|W(_{S})|2d_{S}$.

$+4h^{2-1} \delta\int_{0}^{S}||V(S)||2*ds$

Using Gronwall’s lemma, we obtain that$|W(t)|^{2} \leq ch2||V||^{2}L^{2}(0,s;v’)e\int 0)||2||Y(S)||^{2}+1)\psi\frac{6}{4}s_{(||}Y_{h(s}+(|Yh(S)|+|Y(S)|)2ds$

for all $t\in[0, S]$ . Hence, $Y_{h}arrow Y$ strongly in $C([0, S];\mathcal{H})$ as$harrow \mathrm{O}$.

Step 2.
strongly inrewrite the problem (3.2) in the form$H^{1}(0, S, \mathcal{V}’)\cap C([0, s])\mathcal{H})\cap L^{2}(0, s;\mathcal{V})$$\frac{Y_{h}-Y}{h}arrow Z$$\frac{d}{dt}\frac{Y_{h}-Y}{h}+A\frac{Y_{h}-Y}{h}-\frac{F(Y_{h})-F(Y)}{h}=V(t)$

(3.4)

,

as

$harrow \mathrm{O}$. We$0<t\leq S$ ,$\{$$\frac{Y_{h}-Y}{h}(0)=0$.

On the other hand, we consider the linear problem (3.1). From (a.i), (a.ii), (f.i),(f.ii), and (f.iii), we can easily verify that (3.1) possesses a unique weak solution $Z\in$
on $[0, S]-$ (cf. [4, Chap. XVIII, Theorem 2]).
Define $F_{h}’= \int_{0}^{1}F/(Y+\theta(Y_{h}-Y))d\theta$ . Then $W= \frac{Y_{h}-Y}{h}-Z$ satisfies$H^{1}(0, S, \mathcal{V}^{J})\cap C([0, S];\mathcal{H})\cap L^{2}(0, s;\mathcal{V})$$\frac{d\overline{W}(t)}{dt}+A\overline{W}(t)-F_{h};\overline{W}(t)=(F_{h^{-}}’F’)\mathrm{o}Z(t)$

(3.5)

,

$0<t\leq S$ ,$\{$$\overline{W}(0)=0$.

Taking the scalar product of the equation of (3.5) with$\overline{W}$, we obtain that$\frac{1}{2}\frac{d}{dt}|\overline{W}(t)|^{2}+\langle A\overline{W}(t), \overline{W}(t)\rangle$$=\langle F_{h}’\overline{W}(t), \overline{W}(t)\rangle+\langle(F_{h^{-}}’F_{0}/)Z(t), \overline{W}(t)\rangle$.

$\leq\frac{\delta}{2}||\overline{W}(t)||^{2}+(||Y(t)||^{2}+||Y_{h}(t)-Y(t)||^{2}+1)\mu(|Yh|^{2}+|Y|^{2})|\overline{W}(t)|^{2}$$+ \frac{4}{\delta}||(F_{h}’-F^{l})\mathrm{o}(zt)||_{*}^{2}$

where
(3.6)

$\mu$

,

: $[0, \infty)arrow[0, \infty)$ is some increasing continuous function. Therefore,$| \overline{W}(t)|^{2}+\delta\int_{0}^{t}||\overline{W}(s)||^{2}ds$$\leq\int_{0}^{t}(||Y(s)||^{2}+||Y_{h}(S)||^{2}+1)\mu(|Yh|^{2}+|Y|^{2})|\overline{W}(S)|^{2}ds$$+ \frac{8}{\delta}||(F_{h^{-}}^{J}F’)\mathrm{o}(zt)||_{L^{2}(0,S\mathcal{V}’)}^{2};$.

143
From (f.iii), we have $||F_{h}’Z(t)||_{*}\leq C||Z(t)||,$ $t\in[0, S]$ . Since $Y_{h}arrow Y$ strongly infollows from (f.iv) thatstrongly in$F_{h}’Z(t)arrow F_{0}’z(t)$$\mathcal{V}’\mathrm{a}.\mathrm{e}.$$\mathcal{H}$

, it

.

By the dominated convergence theorem, we haveas$||(F_{h};-F’)0(Zt)||_{L^{2}}^{2}(0,S;\mathcal{V}’)arrow 0$

Using Gronwall’s lemma, it follows from (3.6) that.
$H^{1}(0, S;\mathcal{V}’)\cap C([0, S];\mathcal{H})\cap L^{2}(0, s;\mathcal{V})$$\frac{Y_{h}-Y}{h}$$harrow \mathrm{O}$.

is strongly convergent to $Z$ in$\square$With the aid of this proposition, we can easily show the first order necessary condition.

Theorem 3.2.
be an optimal control of (P) andbe the optimal state, that is.
is the solution to (E) with the control
Then, there exists a unique solution $P\in L^{2}(0, s;\mathcal{V})\cap C([0, S];\mathcal{H})\cap H^{1}(0, S;\mathcal{V}’)$ to thelinear problem$Let\overline{U}$$let\overline{Y}\in L^{2}(0, s;v)\cap c([\mathrm{o}, S];\mathcal{H})\mathrm{n}$$\overline{Y}$$H^{1}(0, S, \mathcal{V}’)$$\overline{U}(t)$$- \frac{dP}{dt}+AP-F’(\overline{Y})^{*}P=D^{*}\Lambda(D\overline{Y}-Yd)$

(3.7)

,

$0\leq t<S$ ,$\{$$P(S)=0$in$\mathcal{V}’$, where A :$\mathcal{V}arrow \mathcal{V}’$is a canonical isomorphism; moreover,

for all$\int_{0}^{S}\langle\Lambda P+\gamma\overline{U}, V-\overline{U}\rangle \mathcal{V}\prime dt\geq 0$

Proof. Since$J$

is G\^ateaux differentiable at$J’(\overline{U})(V-\overline{U})\geq 0$$\overline{U}$and$\mathcal{U}_{ad}$for all$V\in \mathcal{U}_{ad}$.

is convex, it is seen that
$V\in \mathcal{U}_{ad}$.

On the other hand, we verify that
(3.8)$J’( \overline{U})(V-\overline{U})=\int_{0}^{S}\langle DY(\overline{U})-Y_{d}, Dz\rangle vdt+\gamma\int_{0}^{S}\langle\overline{U}, V-\overline{U}\rangle_{v}ldt$

with $Z=Y’(\overline{U})(V-\overline{U})$ . Let $P$ be the unique solution of (3.7) in $H^{1}(0, S;v^{;})\cap$. Fkom (a.i), (a.ii), (f.i), (f.ii), and (f.iii), we can guarantee that$P$
such a solution exists (cf. [4, Chap. XVIII, Theorem 2]). Thus, in view of Proposition
3.1 the first intergal in the right hand side of (3.8) is shown to be$C([0, S];\mathcal{H})\cap L^{2}(0, s;\mathcal{V})$$\int_{0}^{S}\langle DY(\overline{U})-Yd, Dz\rangle_{\mathcal{V}}dt=\int_{0}^{S}\langle D^{*}\Lambda(DY(\overline{U})-Y_{d}), z\rangle dt$$= \int_{0}^{S}\langle-\frac{dP}{dt}+AP-F’(\overline{Y})P, Z*\rangle dt=\int_{0}^{S}\langle P, \frac{dZ}{dt}+AZ-F^{J}(\overline{Y})Z\rangle dt$$= \int_{0}^{S}\langle\Lambda P, V-\overline{U}\rangle v\prime dt$.

144Hence,$\int_{0}^{S}\langle\Lambda P+\gamma\overline{U}, V-\overline{U}\rangle_{\mathcal{V}}Jdt\geq 0$,

for all$V\in \mathcal{U}_{ad}$.

$\square$Remark. Note that our result covers that of $[8, 9]$ when the sensitivity function
is linear function of $\chi(\rho)=b\rho$ ( being a positive constant). Furthermore, since all, our result is also
assumptions of our abstract result are satisfied whenapplied in this case.
$\chi(\rho)$$b$$\rho,$$\chi(\rho)=\frac{b\rho}{1+\rho}$

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$‘\prime optima\iota$$\mathrm{D}\mathrm{u}\mathrm{n}\mathrm{o}\mathrm{d}/\mathrm{G}\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{e}\mathrm{r}$