Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions

TL;DR

This paper investigates the sensitivity of optimal control problems governed by nonlinear evolution inclusions, establishing well-posedness and continuity properties, with an application to nonlinear parabolic systems.

Contribution

It provides a general framework for analyzing the solution set and sensitivity of nonlinear evolution inclusion-based control problems, including continuity of solutions and value functions.

Findings

Solution set nonemptiness and continuous selection established

Hadamard well-posedness of the control problem proved

Continuity of the optimal solution multifunction demonstrated

Abstract

We consider a nonlinear optimal control problem governed by a nonlinear evolution inclusion and depending on a parameter $\lambda$. First we examine the dynamics of the problem and establish the nonemptiness of the solution set and produce continuous selections of the solution multifunction $\xi\mapsto S(\xi)$ ($\xi$ being the initial condition). These results are proved in a very general framework and are of independent interest as results about evolution inclusions. Then we use them to study the sensitivity properties of the optimal control problem. We show that we have Hadamard well-posedness (continuity of the value function) and we establish the continuity properties of the optimal multifunction. Finally we present an application on a nonlinear parabolic distributed parameter system.

Equations

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}J(x,u,\lambda)=\int^{b}_{0}L(t,x(t),\lambda)dt+\int^{b}_{0}H(t,u(t),\lambda)dt+\hat{\psi}(\xi,x(b),\lambda)\rightarrow\inf=\\ \hskip 284.52756ptm(\xi,\lambda),\\ -x^{\prime}(t)\in A_{\lambda}(t,x(t))+F(t,x(t),\lambda)+G(t,u(t),\lambda)\ \mbox{for almost all}\ t\in T,\\ x(0)=\xi,u(t)\in U(t,\lambda)\ \mbox{for almost all}\ t\in T,\lambda\in E.\end{array}\right\}$$

$$A_{\lambda }:T\times X\to 2^{X^{\ast }}\text{mbox}forevery\lambda >0,F:T\times H\times E\to 2^H\setminus \{\varnothing \},G:T\times Y\times E\to 2^H\setminus \{\varnothing \}$$

$${⟨\cdot ,\cdot ⟩|}_{H\times X}=(\cdot ,\cdot ).$$

$$|\cdot |⩽\beta ||\cdot ||.$$

$$W_p(0,b)=\{x\in L^p(T,X):x^{\prime }\in L^{p^{\prime }}(T,X^{\ast })\}\left(\frac{1}{p}+\frac{1}{p^{\prime }}=1\right).$$

$$W_p(0,b)\subseteq A{C}^{1,p^{\prime }}(T,X^{\ast })=W^{1,p^{\prime }}((0,b),X^{\ast }).$$

$$||x|{|}_{W_p}=||x|{|}_{L^p(T,X)}+||x^{\prime }|{|}_{L^{p^{\prime }}(T,X^{\ast })}\text{mbox}forallx\in W_p(0,b),$$

$$W_p(0,b)\hookrightarrow C(T,H),$$

$$W_p(0,b)\hookrightarrow L^p(T,H)\text{mbox}compactly.$$

$$\frac{d}{dt}(x(t),y(t))=⟨x^{\prime }(t),y(t)⟩+⟨x(t),y^{\prime }(t)⟩\text{mbox}foralmostallt\in T.$$

$$L^p(T,X{)}^{\ast }=L^{p^{\prime }}(T,X^{\ast })$$

$$P_{f(c)}(V)=\{C\subseteq V:C\text{mbox}isnonempty,closed,(convex)\}$$

$$P_{(w)k(c)}(V)=\{C\subseteq V:C\text{mbox}isnonempty,(weakly-)compact,(convex)\}.$$

$$\mathrm{Gr}F=\{(\omega ,v)\in \Omega \times V:v\in F(\omega )\}.$$

$$\omega \mapsto d(y,F(\omega ))=\inf [||y-v|{|}_V:v\in F(\omega )]$$

$$S_F^P=\{f\in L^p(\Omega ,Y):f(\omega )\in F(\omega )\mu \text{mbox}-a.e.\}.$$

$${\chi }_A{f}_1+{\chi }_{\Omega \backslash A}{f}_2\in S_F^P.$$

$$|D|=\sup [||v|{|}_V:v\in D]$$

$$\sigma (v^{\ast },D)=\sup [{⟨v^{\ast },v⟩}_V:v\in D]\text{mbox}forall{v}^{\ast }\in V^{\ast }.$$

$$G^{+}(U)=\{z\in Z:G(z)\subseteq U\}\text{mbox}respectively{G}^{-}(U)=\{z\in Z:G(z)\cap U\ne \varnothing \}$$

$$h(E,M)=\max \left[\underset{e\in E}{\sup }d(e,M),\underset{m\in M}{\sup }d(m,E)\right].$$

$$h(E,M)=\sup [|\sigma (v^{\ast },E)-\sigma (v^{\ast },M)|:v^{\ast }\in V^{\ast },||v^{\ast }|{|}_{V^{\ast }}⩽1]$$

$$K_{seq}(\tau )-\underset{n\to \infty }{\mathrm{liminf}}{E}_n=\{y\in W:y=\tau -\underset{n\to \infty }{\lim }{y}_n,y_n\in E_n\text{mbox}foralln\in \mathbb{N}\},$$

$$K_{seq}(\tau )-\underset{n\to \infty }{\mathrm{limsup}}{E}_n=\{y\in W:y=\tau -\underset{n\to \infty }{\lim }{y}_{n_k},y_{n_k}\in E_{n_k},n_1<n_2<\dots <n_k<\dots \}.$$

$$y^{\prime }+a_n(y)\ni h^{\ast }(n\in \mathbb{N})\text{mbox}and{y}^{\prime }+a(y)\ni h$$

$$e_n(h^{\ast })\stackrel{w}{\to }e(h^{\ast })\text{mbox}in{W}_p(0,b).$$

$$((h^{\ast },f))={\int }_0^b⟨h^{\ast }(t),f(t)⟩dt\text{mbox}forall{h}^{\ast }\in L^{p^{\prime }}(T,X^{\ast }),\text{mbox}allf\in L^p(T,X).$$

$$\frac{\inf [{⟨v^{\ast },v⟩}_V:v^{\ast }\in A(v)]}{||v|{|}_V}\to +\infty \text{mbox}as||v|{|}_V\to +\infty .$$

$$\left\{\begin{array}[]{l}-x^{\prime}(t)\in A(t,x(t))+E(t,x(t))\ \mbox{for almost all}\ t\in T,\\ x(0)=\xi.\end{array}\right\}$$

$$||h|{|}_{\ast }⩽a_1(t)+c_1||x|{|}^{p-1}$$

$$⟨h^{\ast },x⟩⩾c_2||x|{|}^p-a_2(t),$$

$$A(x)=-\mathrm{div}\partial \phi (Dx)-\mathrm{div}\xi (Dx)$$

Full text

aa 2010 AMS Subject Classification: Primary 49J24, 49K20, Secondary 34G20

1. Introduction

One of the important problems in optimal control theory, is the study of the variations of the set of optimal state-control pairs and of the value of the problem, when we perturb the dynamics, the cost functional and the initial condition of the problem. Such a sensitivity analysis (also known in the literature as “variational analysis”) is important because it gives information about the tolerances which are permitted in the specification of the mathematical models, it suggests ways to solve parametric problems and also can be useful in the computational analysis of the problem. For infinite dimensional systems (distributed parameter systems), such investigations were conducted by Buttazzo and Dal Maso [8], Denkowski and Migorski [13], Ito and Kunisch [24], Papageorgiou [31] (linear systems), Papageorgiou [30], Sokolowski [38] (semilinear systems) and Hu and Papageorgiou [23], Papageorgiou [32, 33] (nonlinear systems). We also mention the books of Buttazzo [7], Dontchev and Zolezzi [17], Ito and Kunisch [25], Sokolowski and Zolezio [39] (the latter for shape optimization problems). In this paper we conduct such an analysis for a very general class of systems driven by nonmonotone evolution inclusions.

So, let $\displaystyle T=[0,b]$ be the time interval and $\displaystyle(X,H,X^{*})$ an evolution triple of spaces (see Section 2). We assume that $\displaystyle X\hookrightarrow H$ compactly. The space of controls is modelled by a separable reflexive Banach space $\displaystyle Y$ and $\displaystyle E$ is a compact metric space and corresponds to the parameter space. As we have already mentioned, we consider systems monitored by evolution inclusions. These inclusions represent a way to model systems with deterministic uncertainties, see the books of Aubin and Frankowska [2], Fattorini [18], and Roubicek [37].

The problem under consideration is the following:

[TABLE]

In this problem

[TABLE]

and the precise conditions on them will be given in Section 4. For every initial state $\displaystyle\xi\in H$ and every parameter $\displaystyle\lambda\in E$, we denote the set of admissible state-control pairs (that is, pairs $\displaystyle(x,u)$ which satisfy the dynamics and the constraints of problem (5)) by $\displaystyle Q(\xi,\lambda)$. We investigate the dependence of $\displaystyle Q(\xi,\lambda)$ on the two variables $\displaystyle(\xi,\lambda)\in H\times E$. Also, $\displaystyle\Sigma(\xi,\lambda)$ denotes the set of optimal state-control pairs (that is, $\displaystyle(x^{*},u^{*})\in\Sigma(\xi,\lambda)$ such that $\displaystyle J(x^{*},u^{*},\xi,\lambda)=m(\xi,\lambda)$). So, $\displaystyle\Sigma(\xi,\lambda)\subseteq Q(\xi,\lambda)$. We establish the nonemptiness of the set $\displaystyle\Sigma(\xi,\lambda)$ and examine the continuity properties of the value function $\displaystyle(\xi,\lambda)\mapsto m(\xi,\lambda)$ and of the multifunction $\displaystyle(\xi,\lambda)\mapsto\Sigma(\xi,\lambda)$.

The nonemptiness and other continuity and structural properties of the set $\displaystyle Q(\xi,\lambda)$ are consequences of general results about evolution inclusions which we prove in Section 3 and which are of independent interest. The class of evolution inclusions considered in Section 3 is more general than the classes studied by Chen, Wang and Zu [11], Denkowski, Migorski and Papageorgiou [16], Liu [28], Papageorgiou and Kyritsi [34].

In the next section, for the convenience of the reader, we review the main mathematical tools which we will need in this paper.

2. Mathematical Background

Suppose that $\displaystyle V$ and $\displaystyle Z$ are Banach spaces and assume that $\displaystyle V$ is embedded continuously and densely into $\displaystyle Z$ (denoted by $\displaystyle V\hookrightarrow Z$). Then it is easy to check that

• •

$\displaystyle Z^{*}$ is embedded continuously into $\displaystyle V^{*}$;

• •

if $\displaystyle V$ is reflexive, then $\displaystyle Z^{*}\hookrightarrow V^{*}$.

Having this observation in mind, we can introduce the notion of evolution triple of spaces, which is central in the class of evolution equations considered here.

Definition 1**.**

A triple $\displaystyle(X,H,X^{*})$ of spaces is said to be an “evolution triple” (or “Gelfand triple” or “spaces in normal position”), if the following hold:

• (a)

$\displaystyle X$* is a separable reflexive Banach space and $\displaystyle X^{*}$ is its topological dual;*

• (b)

$\displaystyle H$* is a separable Hilbert space identified with its dual $\displaystyle H^{*}=H$ (pivot space);*

• (c)

$\displaystyle X\hookrightarrow H$.

According to the remark made in the beginning of this section, we also have $\displaystyle H^{*}=H\hookrightarrow X^{*}$. In this paper we also assume that the embedding of $\displaystyle X$ into $\displaystyle H$ is compact. Hence by Schauder’s theorem (see, for example, Gasinski and Papageorgiou [20, Theorem 3.1.22, p. 275]), so is the embedding of $\displaystyle H^{*}=H$ into $\displaystyle X^{*}$. In what follows, by $\displaystyle||\cdot||$ (resp. $\displaystyle|\cdot|,\ ||\cdot||_{*}$) we denote the norm of the space $\displaystyle X$ (resp. $\displaystyle H,X^{*}$). By $\displaystyle\left\langle\cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $\displaystyle(X^{*},X)$ and by $\displaystyle(\cdot,\cdot)$ the inner product of the Hilbert space $\displaystyle H$. We know that

[TABLE]

Also, let $\displaystyle\beta>0$ be such that

[TABLE]

We introduce the following space which has a central role in the study of the evolution inclusions. So, let $\displaystyle 1<p<\infty$ and set

[TABLE]

In this definition the derivative of $\displaystyle x$ is understood in the sense of vectorial distributions (weak derivative). In fact, if we view $\displaystyle x$ as an $\displaystyle X^{*}$-valued function, then $\displaystyle x(\cdot)$ is absolutely continuous, hence strongly differentiable almost everywhere. Therefore

[TABLE]

The space $\displaystyle W_{p}(0,b)$, equipped with the norm

[TABLE]

becomes a separable reflexive Banach space. We know that

[TABLE]

The following integration by parts formula is very helpful:

Proposition 2**.**

If $\displaystyle x,y\in W_{p}(0,b)$, then $\displaystyle t\mapsto(x(t),y(t))$ is absolutely continuous and

[TABLE]

We know that for all $\displaystyle 1\leqslant p<\infty$,

[TABLE]

with $\displaystyle p^{\prime}=+\infty$ if $\displaystyle p=1$ (see Gasinski and Papageorgiou [20, Theorem 2.2.9, p. 129]).

Now, let $\displaystyle(\Omega,\Sigma)$ be a measurable space and $\displaystyle V$ a separable Banach space. We introduce the following hyperspaces:

[TABLE]

Given a multifunction $\displaystyle F:\Omega\rightarrow 2^{V}\backslash\{\emptyset\}$, the “graph” of $\displaystyle F$ is the set

[TABLE]

We say that $\displaystyle F(\cdot)$ is “graph measurable” if $\displaystyle{\rm Gr}\,F\in\Sigma\times B(V)$ with $\displaystyle B(V)$ being the Borel $\displaystyle\sigma$-field of $\displaystyle V$. If $\displaystyle\mu(\cdot)$ is a $\displaystyle\sigma$-finite measure on $\displaystyle\Sigma$ and $\displaystyle F:\Omega\rightarrow 2^{V}\backslash\{\emptyset\}$ is graph measurable, then the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [22, Theorem 2.14, p. 158]) implies that $\displaystyle F(\cdot)$ admits a measurable selection, that is, there exists a $\displaystyle\Sigma$-measurable function $\displaystyle f:\Omega\rightarrow V$ such that $\displaystyle f(\omega)\in F(u)$ $\displaystyle\mu$-almost everywhere. In fact, there is a whole sequence $\displaystyle\{f_{n}\}_{n\geqslant 1}$ of such measurable selections such that $\displaystyle F(\omega)\subseteq\overline{\{f_{n}(\omega)\}}$ $\displaystyle\mu$-almost everywhere (see Hu and Papageorgiou [22, Proposition 2.17, p. 159]). Moreover, the above results are valid if $\displaystyle V$ is only a Souslin space. Recall that a Souslin space need not be metrizable (see Gasinski and Papageorgiou [21, p. 232]). A multifunction $\displaystyle F:\Omega\ rightarrowP_{f}(V)$ is said to be “measurable” if for all $\displaystyle y\in V$, the function

[TABLE]

is $\displaystyle\Sigma$-measurable. A multifunction $\displaystyle F:\Omega\rightarrow P_{f}(V)$ which is measurable is also graph measurable. The converse is true if ($\displaystyle\Omega,\Sigma$) admits a complete $\displaystyle\sigma$-finite measure $\displaystyle\mu$. If $\displaystyle(\Omega,\Sigma,\mu)$ is a $\displaystyle\sigma$-finite measure space and $\displaystyle F:\Omega\rightarrow 2^{V}\backslash\{\emptyset\}$ is a multifunction, then for $\displaystyle 1\leqslant p\leqslant\infty$ we introduce the set

[TABLE]

Evidently, $\displaystyle S^{P}_{F}\neq\emptyset$ if and only if $\displaystyle\omega\mapsto\inf[||v||_{V}:v\in F(\omega)]$ belongs to $\displaystyle L^{p}(\Omega)$. Moreover, the set $\displaystyle S^{P}_{F}$ is “decomposable”, that is, if $\displaystyle(A,f_{1},f_{2})\in\Sigma\times S^{P}_{F}\times S^{P}_{F}$, then

[TABLE]

Here, for $\displaystyle C\in\Sigma$, by $\displaystyle\chi_{C}$ we denote the characteristic function of the set $\displaystyle C\in\Sigma$.

For every $\displaystyle D\subseteq\Sigma,\ D\neq\emptyset$, we define

[TABLE]

Here, $\displaystyle\left\langle\cdot,\cdot\right\rangle_{V}$ denotes the duality brackets of the pair $\displaystyle(V^{*},V)$. The function $\displaystyle\sigma(\cdot,D):V^{*}\rightarrow\bar{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}$ is known as the “support function” of $\displaystyle D$.

Let $\displaystyle Z,W$ be Hausdorff topological spaces. We say that a multifunction $\displaystyle G:Z\rightarrow 2^{W}\backslash\{\emptyset\}$ is “upper semicontinuous” (usc for short), respectively “lower semicontinuous” (lsc for short), if for all $\displaystyle U\subseteq W$ open, the set

[TABLE]

is open in $\displaystyle Z$. If $\displaystyle G(\cdot)$ is both usc and lsc, then we say that $\displaystyle G(\cdot)$ is continuous. On a Hausdorff topological space $\displaystyle(W,\tau)$ ($\displaystyle\tau$ being the Hausdorff topology), we can define a new topology $\displaystyle\tau_{seq}$ whose closed sets are the sequentially $\displaystyle\tau$-closed sets. Then topological properties with respect to this topology have the prefix “sequential”. Note that $\displaystyle\tau\subseteq\tau_{seq}$ and the two are equal, if $\displaystyle\tau$ is first countable (see Buttazzo [7, p. 9] and Gasinski and Papageorgiou [21, p. 808]). We say that $\displaystyle G:Z\rightarrow 2^{W}\backslash\{\emptyset\}$ is “closed”, if the graph $\displaystyle{\rm Gr}\,G\subseteq Z\times W$ is closed.

For any Banach space $\displaystyle V$, on $\displaystyle P_{f}(V)$ we can define a generalized metric, known as the “Hausdorff metric”, by setting

[TABLE]

Recall that $\displaystyle(P_{f}(V),h)$ is a complete metric space (see Hu and Papageorgiou [22, p. 6]). If $\displaystyle Z$ is a Hausdorff topological space, a multifunction $\displaystyle G:Z\rightarrow P_{f}(V)$ is said to be “$\displaystyle h$-continuous”, if it is continuous from $\displaystyle Z$ into $\displaystyle(P_{f}(V),h)$.

Also, if $\displaystyle E,M\subseteq V$ are nonempty, bounded, closed and convex subsets, then

[TABLE]

(Hörmander’s formula).

Let $\displaystyle(W,\tau)$ be a Hausdorff topological space with topology $\displaystyle\tau$ and let $\displaystyle\{E_{n}\}_{n\geqslant 1}\subseteq 2^{W}\backslash\{\emptyset\}$. We define

[TABLE]

Sometimes we drop the $\displaystyle K_{seq}$-symbol and simply write $\displaystyle\tau-\limsup_{n\rightarrow\infty}E_{n}$ and $\displaystyle\tau-\liminf\limits_{n\rightarrow\infty}E_{n}$.

Returning to the setting of an evolution triple, we consider a sequence of multivalued maps $\displaystyle a_{n},a:L^{p}(T,X)\rightarrow 2^{L^{p^{\prime}}(T,X^{*})}\backslash\{\emptyset\}$ ($\displaystyle n\in\mathbb{N}$) such that for every $\displaystyle h^{*}\in L^{p^{\prime}}(T,X^{*})$ the inclusions

[TABLE]

have unique solutions $\displaystyle e_{n}(h^{*}),e(h^{*})\in W_{p}(0,b)$.

We say that $\displaystyle\frac{d}{dt}+a_{n}$ “PG-converges” to $\displaystyle\frac{d}{dt}+a$ (denoted by $\displaystyle\frac{d}{dt}+a_{n}\xrightarrow{PG}\frac{d}{dt}+a$ as $\displaystyle n\rightarrow\infty$), if for every $\displaystyle h^{*}\in L^{p^{\prime}}(T,X^{*})$ we have

[TABLE]

In what follows, by $\displaystyle X_{w}$ (respectively $\displaystyle H_{w},X^{*}_{w}$) we denote the space $\displaystyle X$ (respectively $\displaystyle H,X^{*}$) furnished with the weak topology. Also, by $\displaystyle|\cdot|_{1}$ we denote the Lebesgue measure on $\displaystyle\mathbb{R}$ and by $\displaystyle((\cdot,\cdot))$ we denote the duality brackets for the pair $\displaystyle(L^{p^{\prime}}(T,X^{*}),L^{p}(T,X))$. So, we have

[TABLE]

Next, let us recall some useful facts from the theory of nonlinear operators of monotone type.

So, let $\displaystyle V$ be a reflexive Banach space, $\displaystyle L:D(L)\subseteq V\rightarrow V^{*}$ a linear maximal monotone operator and $\displaystyle a:V\rightarrow 2^{V^{*}}$. We say that $\displaystyle a(\cdot)$ is “$\displaystyle L$-pseudomonotone” if the following conditions hold:

• (a)

For every $\displaystyle v\in V,a(v)\in P_{wkc}(V^{*})$.

• (b)

$\displaystyle a(\cdot)$ is bounded (that is, maps bounded sets to bounded sets).

• (c)

If $\displaystyle\{v_{n}\}_{n\geqslant 1}\subseteq D(L),\ v_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}v\in D(L)$ in $\displaystyle V$, $\displaystyle L(v_{n})\stackrel{{\scriptstyle w}}{{\rightarrow}}L(v)$ in $\displaystyle V^{*}$, $\displaystyle v^{*}_{n}\in a(v_{n})$ for all $\displaystyle n\in\mathbb{N}$, $\displaystyle v^{*}_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}v^{*}$ in $\displaystyle X^{*}$ and $\displaystyle\limsup\limits_{n\rightarrow\infty}\left\langle v^{*}_{n},v_{n}-v\right\rangle_{V}\leqslant 0,$ then $\displaystyle v^{*}\in a(v)$ and $\displaystyle\left\langle v^{*}_{n},v_{n}\right\rangle_{V}\rightarrow\left\langle v^{*},v\right\rangle_{V}$

Such maps have nice surjectivity properties. The next result is due to Papageorgiou, Papalini and Renzacci [35] and it extends an earlier single-valued result of Lions [27, Theorem 1.2, p. 319].

Proposition 3**.**

Assume that $\displaystyle V$ is a reflexive Banach space which is strictly convex, $\displaystyle L:D(L)\subseteq V\rightarrow V^{*}$ is a linear maximal monotone operator and $\displaystyle A:V\rightarrow 2^{V^{*}}$ is $\displaystyle L$-pseudomonotone and strongly coercive, that is,

[TABLE]

Then $\displaystyle R(L+V)=V^{*}$ (that is, $\displaystyle L+V$ is surjective).

In the next section we obtain some results about a general class of evolution inclusions, which will help us study problem (5) (see Section 4).

3. Nonlinear Evolution Inclusions

Let $\displaystyle T=[0,b]$ and let ($\displaystyle X,H,X^{*}$) be an evolution triple with $\displaystyle X\hookrightarrow H$ compactly (see Definition 1). In this section we deal with the following evolution inclusion:

[TABLE]

The hypotheses on the data of (9) are the following:

$\displaystyle H(A)_{1}:\quad A:T\times X\rightarrow 2^{X^{*}}$ is a map such that

• (i)

for all $\displaystyle x\in X,\ t\mapsto A(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T,\ {\rm Gr}\,A(t,\cdot)$ is sequentially closed in $\displaystyle X_{w}\times X^{*}_{w}$ and $\displaystyle x\mapsto A(t,x)$ is pseudomonotone;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$ and all $\displaystyle h^{*}\in A(t,x)$, we have

[TABLE]

with $\displaystyle 2\leqslant p,\ a_{1}\in L^{p^{\prime}}(T)$ and $\displaystyle c_{1}>0$;

• (iv)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$ and all $\displaystyle h^{*}\in A(t,x)$, we have

[TABLE]

with $\displaystyle c_{2}>0,\ a_{2}\in L^{1}(T)$.

Remark 1**.**

If $\displaystyle A(\cdot,\cdot)$ is single-valued, then in hypothesis $\displaystyle H(A)(ii)$ we can drop the condition on the graph of $\displaystyle{\rm Gr}\,A(t,\cdot)$ and only assume that for almost all $\displaystyle t\in T,\ x\mapsto A(t,x)$ is pseudomonotone. Similarly, if for almost all $\displaystyle t\in T,A(t,\cdot)$ is maximal monotone. An example of where the condition on the graph of $\displaystyle A(t,\cdot)$ is satisfied is the following. For simplicity we drop the $\displaystyle t$-dependence

[TABLE]

where $\displaystyle\varphi:L^{p}(\Omega,\mathbb{R}^{N})\rightarrow\mathbb{R}$ is continuous, convex and $\displaystyle\xi:L^{p}(\Omega,\mathbb{R}^{N})\rightarrow\mathbb{R}$ is continuous and $\displaystyle|\xi(y)|\leqslant\hat{c}(1+|y|^{\tau-1})$ for all $\displaystyle y\in\mathbb{R}^{N}$, some $\displaystyle\hat{c}>0$ and with $\displaystyle 1\leqslant\tau<p$. Then recalling that $\displaystyle W^{1,p}(\Omega)\hookrightarrow W^{1,\tau}(\Omega)$ compactly (see Zeidler [41, p. 1026]), we easily see that $\displaystyle{\rm Gr}\,A$ is sequentially closed in $\displaystyle W^{1,p}(\Omega)_{w}\times W^{1,p}(\Omega)^{*}_{w}$.

$\displaystyle H(F)_{1}:$ $\displaystyle F:T\times H\rightarrow P_{{f_{c}}}(H)$ is a multifunction such that

• (i)

for all $\displaystyle x\in H$, $\displaystyle t\mapsto F(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T,\ {\rm Gr}\,F(t,\cdot)$ is sequentially closed in $\displaystyle H\times H_{w}$;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$ and all $\displaystyle h\in F(t,x)$

[TABLE]

with $\displaystyle a_{3}\in L^{2}(T),c_{3}>0$ and if $\displaystyle p=2$, then $\displaystyle\beta^{2}c_{3}<c_{2}$ (see (6)).

By a solution of problem (9) we understand a function $\displaystyle x\in W_{p}(0,b)$ such that

[TABLE]

with $\displaystyle h^{*}\in L^{p^{\prime}}(T,X^{*}),f\in L^{2}(T,H)$ such that

[TABLE]

By $\displaystyle S(\xi)$ we denote the set of solutions of problem (9). In the sequel we investigate the structure of $\displaystyle S(\xi)$.

Consider the multivalued map $\displaystyle a:L^{p}(T,X)\rightarrow 2^{L^{p^{\prime}}(T,X^{*})}$ defined by

[TABLE]

Note that $\displaystyle a(\cdot)$ has values in $\displaystyle P_{wkc}(L^{p^{\prime}}(T,X^{*}))$ (see hypotheses $\displaystyle H(A)(i),(iii)$ and use the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [22, Theorem 2.15, p. 518])).

Lemma 4**.**

If hypotheses $\displaystyle H(A)_{1}$ hold, $\displaystyle x_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}x$ in $\displaystyle W_{p}(0,b)$, $\displaystyle x_{n}(t)\stackrel{{\scriptstyle w}}{{\rightarrow}}x(t)$ in $\displaystyle X$ for almost all $\displaystyle t\in T$, $\displaystyle h^{*}_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}h^{*}$ in $\displaystyle L^{p^{\prime}}(T,X^{*})$ and $\displaystyle h^{*}_{n}\in a(x_{n})$ for all $\displaystyle n\in\mathbb{N}$, then $\displaystyle h^{*}\in a(x)$.

Proof.

Let $\displaystyle v\in X$ and consider the function $\displaystyle x\mapsto\sigma(v,A(t,x))$ (see Section 2). We will show that it is sequentially upper semicontinuous. To this end we need to show that given $\displaystyle\lambda\in\mathbb{R}$, the superlevel set

[TABLE]

is sequentially closed in $\displaystyle X_{w}$. So, we consider a sequence $\displaystyle\{x_{n}\}_{n\geqslant 1}\subseteq E_{\lambda}$ and assume that

[TABLE]

Let $\displaystyle h^{*}_{n}\in A(t,x_{n})$ ($\displaystyle n\in\mathbb{N}$) be such that

[TABLE]

(recall $\displaystyle A(t,x_{n})\in P_{wkc}(X^{*})$). Evidently, $\displaystyle\{h^{*}_{n}\}_{n\geqslant 1}\subseteq X^{*}$ is bounded (see hypothesis

$\displaystyle H(A)_{1}(iii)$) and so by passing to a subsequence if necessary, we may assume that

[TABLE]

Then we have

[TABLE]

This proves the upper semicontinuity of the map $\displaystyle x\mapsto\sigma(v,A(t,x))$.

Now let $\displaystyle v\in L^{p}(T,X)$. We have

[TABLE]

∎

Lemma 5**.**

If hypotheses $\displaystyle H(A)_{1}$ hold, then the multivalued map $\displaystyle a:L^{p}(T,X)\rightarrow 2^{L^{p^{\prime}}(T,X^{*})}$ defined by (10) is $\displaystyle L$-pseudomonotone.

Proof.

Suppose $\displaystyle x_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}x$ in $\displaystyle W_{p}(0,b),h^{*}_{n}\in a(x_{n})$ for all $\displaystyle n\in\mathbb{N}$, $\displaystyle h^{*}_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}h^{*}$ in $\displaystyle L^{p^{\prime}}(T,X^{*})$ and

[TABLE]

From (7) we infer that

[TABLE]

We set $\displaystyle\vartheta_{n}(t)=\left\langle h^{*}_{n}(t),x_{n}(t)-x(t)\right\rangle$. Let $\displaystyle N$ be the Lebesgue-null set in $\displaystyle T=[0,b]$ outside of which hypotheses $\displaystyle H(A)_{1}(ii),(iii),(iv)$ hold. Using hypotheses $\displaystyle H(A)_{1}(iii),(iv)$ we have

[TABLE]

We introduce the Lebesgue measurable set $\displaystyle D\subseteq T$ defined by

[TABLE]

Suppose that $\displaystyle|D|_{1}>0$. If $\displaystyle t\in D\cap(T\backslash N)$, then from (15) we see that

[TABLE]

Then it follows from (14) that

[TABLE]

We fix $\displaystyle t\in D\cap(T\backslash N)$ and choose a subsequence $\displaystyle\{n_{k}\}$ of $\displaystyle\{n\}$ (in general this subsequence depends on $\displaystyle t$) such that

[TABLE]

By hypothesis $\displaystyle H(A)_{1}(ii)$, $\displaystyle A(t,\cdot)$ is pseudomonotone and since $\displaystyle t\in D$, we infer that

[TABLE]

a contradiction. So, $\displaystyle|D|_{1}=0$ and we have

[TABLE]

Invoking the extended Fatou’s lemma (see Denkowski, Migorski and Papageorgiou [15, Theorem 2.2.33, p. 145]), we have

[TABLE]

We write

[TABLE]

Note that

[TABLE]

Moreover, from (15) we have

[TABLE]

with $\displaystyle\{\eta_{n}\}_{n\geqslant 1}\subseteq L^{1}(T)$ uniformly integrable. Then

[TABLE]

with $\displaystyle\{\eta^{-}_{n}\}_{n\geqslant 1}\subseteq L^{1}(T)$ uniformly integrable. Using (20) and invoking Vitali’s theorem we infer that

[TABLE]

Then we have

[TABLE]

In addition, from (16) and Lemma 4, we have that

[TABLE]

This proves the $\displaystyle L$-pseudomonotonicity of $\displaystyle a(\cdot)$. ∎

Remark 2**.**

From the above proof it is clear why in the case of a single-valued map $\displaystyle A(t,x)$, in hypothesis $\displaystyle H(A)_{1}(ii)$ we can drop the condition on the graph of $\displaystyle A(t,\cdot)$ and only assume that for almost all $\displaystyle t\in T$ $\displaystyle x\mapsto A(t,x)$ is pseudomonotone. Indeed, in this case, from (21) we have (at least for a subsequence) that

[TABLE]

In the multivalued case, there is no canonical way to identify the pointwise limit of the sequence $\displaystyle\{h^{*}_{n}(t)\}_{n\geqslant 1}\subseteq X^{*}$. If for almost all $\displaystyle t\in T,A(t,\cdot)$ is maximal monotone, then again, we do not need the graph hypothesis on $\displaystyle A(t,\cdot)$. In this case $\displaystyle a(\cdot)$ is also maximal monotone and then the lemma is a consequence of (13) and Lemma 1.3, p. 42 of Barbu [4]. It is worth mentioning that a similar strengthening of the topology in the range space was used by Defranceschi [12], while studying $\displaystyle G$-convergence of multivalued operators.

Without loss of generality, invoking the Troyanski renorming theorem (see Gasinski and Papageorgiou [21, Remark 2.115, p. 241]), we may assume that both $\displaystyle X$ and $\displaystyle X^{*}$ are locally uniformly convex, hence $\displaystyle L^{p}(T,X)$ and $\displaystyle L^{p^{\prime}}(T,X^{*})$ are strictly convex.

We are now ready for the first result concerning the solution set $\displaystyle S(\xi)$.

Theorem 6**.**

If hypotheses $\displaystyle H(A)_{1},H(F)_{1}$ hold and $\displaystyle\xi\in H$, then the solution set $\displaystyle S(\xi)$ is nonempty, weakly compact in $\displaystyle W_{p}(0,b)$ and compact in $\displaystyle C(T,H)$.

Proof.

First suppose that $\displaystyle\xi\in X$. We define

[TABLE]

Evidently, $\displaystyle A_{1}(t,x)$ and $\displaystyle F_{1}(t,x)$ have the same measurability, continuity and growth properties as the multivalued maps $\displaystyle A(t,x)$ and $\displaystyle F(t,x)$. So, we may equivalently consider the following Cauchy problem

[TABLE]

Note that if $\displaystyle x\in W_{p}(0,b)$ is a solution of (22), then $\displaystyle\hat{x}=x-\xi$ is a solution of (9) (when $\displaystyle\xi\in X$, that is, the initial condition is regular). Consider the linear densely defined operator $\displaystyle L:D(L)\subseteq L^{p}(T,X)\rightarrow L^{p^{\prime}}(T,X^{*})$ defined by

[TABLE]

(the evaluation $\displaystyle y(0)=0$, makes sense by virtue of (7)).

Consider the multivalued maps $\displaystyle a_{1},G_{1}:L^{p}(T,X)\rightarrow 2^{L^{p^{\prime}}(T,X^{*})}\backslash\{\emptyset\}$ defined by

[TABLE]

We set $\displaystyle K(x)=a_{1}(x)+G_{1}(x)$ for all $\displaystyle x\in L^{p}(T,X)$. Then

[TABLE]

Claim 1**.**

$\displaystyle K$* is $\displaystyle L$-pseudomonotone.*

Clearly, $\displaystyle K$ has values in $\displaystyle P_{wkc}(L^{p^{\prime}}(T,X^{*}))$ and it is bounded (see hypotheses $\displaystyle H(A)(iii)$, $\displaystyle H(F)(iii)$).

Next, we consider a sequence $\displaystyle\{x_{n}\}_{n\geqslant 1}\subseteq D(L)$ such that

[TABLE]

Then we have

[TABLE]

Hypotheses $\displaystyle H(A)(iii)$ and $\displaystyle H(F)(iii)$ imply that

[TABLE]

So, we may assume (at least for a subsequence) that

[TABLE]

By (25) we have

[TABLE]

Recall that

[TABLE]

By (25), (3), (27) and Proposition 6.6.33 on p. 521 of Papageorgiou and Kyritsi [34], we have

[TABLE]

Since $\displaystyle((f_{n},x_{n}-x))=\int^{b}_{0}(f_{n}(t),x_{n}(t)-x(t))dt\rightarrow 0$, we conclude that $\displaystyle K$ is $\displaystyle L$-pseudomonotone. This proves Claim 1.

Claim 2**.**

$\displaystyle K$* is coercive.*

Let $\displaystyle x\in L^{p}(T,X)$ and $\displaystyle k^{*}\in K(x)$. Then

[TABLE]

We have

[TABLE]

Note that

[TABLE]

Returning to (28) and using (3) we see that

[TABLE]

It follows from (30) that $\displaystyle K$ is coercive. This proves Claim 2.

Now Claims 1 and 2 permit the use of Proposition 3 to find $\displaystyle x\in W_{p}(0,b)$ solving problem (9) when $\displaystyle\xi\in X$.

Next, we remove the restriction $\displaystyle\xi\in X$. So, suppose $\displaystyle\xi\in H$. We can find $\displaystyle\{\xi_{n}\}_{n\geqslant 1}\subseteq X$ such that $\displaystyle\xi_{n}\rightarrow\xi$ in $\displaystyle H$ (recall that $\displaystyle X$ is dense in $\displaystyle H$). From the first part of the proof, we know that we can find $\displaystyle x_{n}\in S(\xi_{n})\subseteq W_{p}(0,b)$ for all $\displaystyle n\in\mathbb{N}$. We have

[TABLE]

It follows that

[TABLE]

We have

[TABLE]

From (31), (3) and hypotheses $\displaystyle H(A)_{1}(iii),H(F)_{1}(iii)$ it follows that

[TABLE]

So, we may assume that

[TABLE]

By (31) we have for all $\displaystyle n\in\mathbb{N}$

[TABLE]

By Proposition 2 we know that

[TABLE]

Hypothesis $\displaystyle H(F)_{1}(iii)$ implies that $\displaystyle\{f_{n}\}_{n\geqslant 1}\subseteq L^{2}(T,H)$ is bounded. Hence

[TABLE]

By hypothesis $\displaystyle H(A)_{1}(iii)$ we see that

[TABLE]

So, we may assume that

[TABLE]

From (33), (36), (37), we see that we can use Lemma 5 and infer that

[TABLE]

As we have already mentioned $\displaystyle\{f_{n}\}_{n\geqslant 1}\subseteq L^{2}(T,H)$ is bounded and so we may assume that

[TABLE]

Using Proposition 6.6.33 on p. 521 of Papageorgiou and Kyritsi [34], we have

[TABLE]

In (31) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (33), (37), (39) to obtain

[TABLE]

So, we have proved that when $\displaystyle\xi\in H$, the solution set $\displaystyle S(\xi)$ is a nonempty subset of $\displaystyle W_{p}(0,b)$.

Next, we will prove the compactness of $\displaystyle S(\xi)$ in $\displaystyle W_{p}(0,b)_{w}$ and in $\displaystyle C(T,H)$. Let $\displaystyle x\in S(\xi)$. For every $\displaystyle t\in T$ we have

[TABLE]

Then let $\displaystyle r_{M}:H\rightarrow H$ be the $\displaystyle M$-radial retraction defined by

[TABLE]

Because of the a priori bound (41), we can replace $\displaystyle F(t,x)$ by

[TABLE]

Note that for all $\displaystyle x\in H,\ t\mapsto\hat{F}(t,x)$ is graph measurable (hence also measurable, see Section 2) and for almost all $\displaystyle t\in T$, $\displaystyle x\mapsto\hat{F}(t,x)$ has a graph which is sequentially closed in $\displaystyle H\times H_{w}$. Moreover, we see that

[TABLE]

We introduce the set

[TABLE]

We consider the following Cauchy problem

[TABLE]

Let $\displaystyle H:C\rightarrow 2^{C(T,H)}$ be the map (in general, multivalued) that assigns to each $\displaystyle f\in C$ the set of solutions of problem (42). It is a consequence of Proposition 3 and Lemma 5, that $\displaystyle H(\cdot)$ has nonempty values.

Claim 3**.**

$\displaystyle H(C)\subseteq C(T,H)$* is compact.*

Let $\displaystyle\{x_{n}\}_{n\geqslant 1}\subseteq H(C)$. Then

[TABLE]

Hence for all $\displaystyle t\in T$, we have

[TABLE]

Also, using hypothesis $\displaystyle H(A)_{1}(iv)$ we have

[TABLE]

It follows from (43) and (45) that

[TABLE]

So, we may assume that

[TABLE]

Passing to the limit as $\displaystyle n\rightarrow\infty$ in (43) and using (46), we obtain

[TABLE]

Also, from (43) we have

[TABLE]

Note that

[TABLE]

Also using Proposition 2, we have

[TABLE]

It follows from (46) that

[TABLE]

Returning to (47), passing to the limit as $\displaystyle n\rightarrow\infty$ and using (48), (49), (50) we obtain

[TABLE]

From the proof of Lemma 5 (see (21)), we know that

[TABLE]

In a similar fashion we also have

[TABLE]

Also, by (8), (3), (46) and Vitali’s theorem, we have

[TABLE]

For every $\displaystyle t\in T$ and every $\displaystyle n\in\mathbb{N}$, using Proposition 2, we have

[TABLE]

However, from the previous parts of the proof it is clear that $\displaystyle S(\xi)\subseteq H(C)$ is weakly closed in $\displaystyle W_{p}(0,b)$ and closed in $\displaystyle C(T,H)$. Therefore we conclude that $\displaystyle S(\xi)$ is weakly compact in $\displaystyle W_{p}(0,b)$ and compact in $\displaystyle C(T,H)$. ∎

Next, we want to produce a continuous selection of the multifunction $\displaystyle\xi\mapsto S(\xi)$ (we refer to Repovš and Semenov [36] for more details about continuous selections of multivalued mappings). Note that $\displaystyle S(\cdot)$ is in general not convex-valued and so the Michael selection theorem (see Hu and Papageorgiou [22, Theorem 4.6, p. 92]) cannot be used. To produce a continuous selection of the solution multifunction $\displaystyle\xi\mapsto S(\xi)$, we need to strengthen the conditions on the multimap $\displaystyle A(t,\cdot)$, in order to guarantee that certain Cauchy problems admit a unique solution.

The new hypotheses on the map $\displaystyle A(t,x)$ are the following:

$\displaystyle H(A)_{2}:$ $\displaystyle A:T\times X\rightarrow 2^{X^{*}}\backslash\{\emptyset\}$ is a multivalued map such that

• (i)

for every $\displaystyle x\in X,\ t\mapsto A(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T,\ x\mapsto A(t,x)$ is maximal monotone;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$ and all $\displaystyle h^{*}\in A(t,x)$, we have

[TABLE]

with $\displaystyle a_{1}\in L^{p^{\prime}}(T),c_{1}>0,2\leqslant p$;

• (iv)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$ and $\displaystyle h^{*}\in A(t,x)$, we have

[TABLE]

with $\displaystyle c_{2}>0,a_{2}\in L^{1}(T)_{+}$.

Remark 3**.**

As we have already mentioned in an earlier remark, since now for almost all $\displaystyle t\in T$, $\displaystyle A(t,\cdot)$ is maximal monotone, we do not need the condition on the graph of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)_{1}(ii)$ and Barbu [4, Lemma 1.3, p. 42]).

Also, we strengthen the condition on the multifunction $\displaystyle F(t,\cdot)$.

$\displaystyle H(F)_{2}:$ $\displaystyle F:T\times H\rightarrow P_{f_{c}}(H)$ is a multifunction such that

• (i)

for every $\displaystyle x\in H$, $\displaystyle t\mapsto F(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$ and all $\displaystyle x,y\in H$ we have

[TABLE]

with $\displaystyle k\in L^{1}(T)_{+}$;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$ and all $\displaystyle h\in F(t,x)$, we have

[TABLE]

with $\displaystyle a_{3}\in L^{2}(T)_{+},c_{3}>0$ and if $\displaystyle p=2$, then $\displaystyle\beta^{2}c_{3}<c_{2}$ (see (6)).

Remark 4**.**

Hypothesis $\displaystyle H(F)_{2}(ii)$ is stronger than condition $\displaystyle H(F)_{1}(ii)$. Indeed, suppose that $\displaystyle H(F)_{2}(ii)$ holds and we have

[TABLE]

By the definition of the Hausdorff metric (see Section 2), we have

[TABLE]

The function $\displaystyle y\mapsto d(y,F(t,x))$ is continuous and convex, hence weakly lower semicontinuous. Therefore by (54) we have

[TABLE]

This proves that condition $\displaystyle H(F)_{1}(ii)$ holds.

So, we can use Theorem 6 and establish that given any $\displaystyle\xi\in H$, the solution set $\displaystyle S(\xi)$ is nonempty, weakly compact in $\displaystyle W_{p}(0,b)$ and compact in $\displaystyle C(T,H)$. The next result extends an earlier result of Cellina and Ornelas [9] for differential inclusions in $\displaystyle\mathbb{R}^{N}$ with $\displaystyle A\equiv 0$.

Proposition 7**.**

If hypotheses $\displaystyle H(A)_{2},H(F)_{2}$ hold, then there exists a continuous map $\displaystyle\vartheta:H\rightarrow C(T,H)$ such that

[TABLE]

Proof.

Consider the following auxiliary Cauchy problem

[TABLE]

This problem has a unique solution $\displaystyle x_{0}(\xi)\in W_{p}(0,b)$ (see Proposition 3 and use the monotonicity of $\displaystyle A(t,\cdot)$ and Proposition 2 to check the uniqueness of this solution).

If $\displaystyle\xi_{1},\xi_{2}\in H$, then

[TABLE]

So, using Proposition 2, we have

[TABLE]

We consider the multifunction $\displaystyle\Gamma_{0}:H\rightarrow P_{wkc}(L^{1}(T,H))$ defined by

[TABLE]

We have

[TABLE]

Also, $\displaystyle\Gamma_{0}(\cdot)$ has decomposable values. So, we can apply the selection theorem of Bressan and Colombo [5] (see also Hu and Papageorgiou [22, Theorem 8.7, p. 245]) and find a continuous map $\displaystyle\gamma_{0}:H\rightarrow L^{1}(T,H)$ such that $\displaystyle\gamma_{0}(\xi)\in\Gamma_{0}(\xi)$ for all $\displaystyle\xi\in H$. Evidently $\displaystyle\gamma_{0}(\xi)\in L^{2}(T,H)$ for all $\displaystyle\xi\in H$.

We consider the following auxiliary Cauchy problem:

[TABLE]

This problem has a unique solution $\displaystyle x_{1}(\xi)\in W_{p}(0,b)$.

By induction we will produce two sequences

[TABLE]

which satisfy:

• (a)

$\displaystyle x_{n}(\xi)\in W_{p}(0,b)$ is the unique solution of the Cauchy problem

[TABLE]

• (b)

$\displaystyle\xi\mapsto\gamma_{n}(\xi)$ is continuous from $\displaystyle H$ into $\displaystyle C(T,H)$;

• (c)

$\displaystyle\gamma_{n}(\xi)(t)\in F(t,x_{n}(\xi)(t))$ for almost all $\displaystyle t\in T$, all $\displaystyle\xi\in H$;

• (d)

$\displaystyle|\gamma_{n}(\xi)(t)-\gamma_{n-1}(\xi)(t)|\leqslant k(t)\beta_{n}(\xi)(t)$ for almost all $\displaystyle t\in T$, all $\displaystyle\xi\in H$, where

[TABLE]

with $\displaystyle\epsilon>0,\eta(\xi)(t)=a_{2}(t)+c_{2}|x_{0}(\xi)(t)|,\tau(t)=\int^{t}_{0}k(s)ds.$

Note that the maps $\displaystyle\xi\mapsto\eta(\xi)$ and $\displaystyle\xi\mapsto\beta_{n}(\xi)$ are continuous from $\displaystyle H$ into $\displaystyle L^{1}(T)$.

So, suppose we have produced $\displaystyle\{x_{k}(\xi)\}^{n}_{k=1}$ and $\displaystyle\{\gamma_{k}(\xi)\}^{n}_{k=1}$ (induction hypothesis). Let $\displaystyle x_{n+1}(\xi)\in W_{p}(0,b)$ be the unique solution of the Cauchy problem

[TABLE]

By (56) and (57) we have

[TABLE]

Using (58) we can write

[TABLE]

By (3) and Lemma A.5, p. 157 of Brezis [6], we infer that

[TABLE]

Using the induction hypothesis (see (c)) and hypothesis $\displaystyle H(F)_{2}(ii)$, we have

[TABLE]

Consider the multifunction $\displaystyle\Gamma_{n+1}:H\rightarrow 2^{L^{1}(T,H)}$ defined by

[TABLE]

By (62) and Lemma 8.3 on p. 239 of Hu and Papageorgiou [22], we have that

[TABLE]

We can apply the selection theorem of Bressan and Colombo [5, Theorem 3] to find a continuous map $\displaystyle\gamma_{n+1}:H\rightarrow L^{1}(T,H)$ such that $\displaystyle\gamma_{n+1}(\xi)\in\overline{\Gamma_{n+1}(\xi)}$ for all $\displaystyle\xi\in H$.

This completes the induction process and we have produced two sequences $\displaystyle\{x_{n}(\xi)\}_{n\geqslant 1}$, $\displaystyle\{\gamma_{n}(\xi)\}_{n\geqslant 1}$ which satisfy properties $\displaystyle(a)\rightarrow(d)$ stated earlier.

From (61) we have

[TABLE]

Recall that $\displaystyle\xi\mapsto\eta(\xi)$ is continuous from $\displaystyle H$ into $\displaystyle L^{1}(H)$ and maps bounded sets to bounded sets. So, it follows from (63) that

[TABLE]

Moreover, from (61) and (63), we have

[TABLE]

Therefore we have

[TABLE]

Evidently, $\displaystyle\xi\mapsto x(\xi)$ is continuous from $\displaystyle H$ into $\displaystyle C(T,H)$, while because of hypothesis $\displaystyle H(F)_{2}(iii)$, we have $\displaystyle\gamma_{n}(\xi)\rightarrow\gamma(\xi)$ in $\displaystyle L^{2}(T,H)$. Let $\displaystyle\hat{x}(\xi)\in W_{p}(0,b)$ be the unique solution of

[TABLE]

As before, exploiting the monotonicity of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)_{2}(ii)$), we have

[TABLE]

So, $\displaystyle x(\xi)\in S(\xi)$ and the map $\displaystyle\vartheta:H\rightarrow C(T,H)$ defined by $\displaystyle\vartheta(\xi)=x(\xi)$ is a continuous selection of the solution multifunction $\displaystyle\xi\mapsto S(\xi)$. ∎

An easy, but useful consequence of Proposition 7 and of its proof, is a parametric version of the Filippov-Gronwall inequality (see Aubin and Cellina [1, Theorem 1, pp. 120-121] and Frankowska [19]) for differential inclusions.

So, we consider the following parametric version of problem (9):

[TABLE]

The parameter space $\displaystyle D$ is a complete metric space. The hypotheses on the parametric vector field $\displaystyle F(t,x,\lambda)$ and the initial condition $\displaystyle\xi(\lambda)$ are the following:

$\displaystyle H(F)^{\prime}_{2}:$ $\displaystyle F:T\times H\times D\rightarrow P_{f_{c}}(H)$ is a multifunction such that

• (i)

for all $\displaystyle(x,\lambda)\in H\times D,\ t\mapsto F(t,x,\lambda)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, all $\displaystyle x,y\in H$, all $\displaystyle\lambda\in D$, we have

[TABLE]

with $\displaystyle k\in L^{1}(T)_{+}$;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$, all $\displaystyle\lambda\in D$ and all $\displaystyle h\in F(t,x)$, we have

[TABLE]

with $\displaystyle a_{3}\in L^{2}(T)_{+},\ c_{3}>0$ and if $\displaystyle p=2$, then $\displaystyle\beta^{2}c_{3}<c_{2}$ (see (6));

• (iv)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$, the multifunction $\displaystyle\lambda\mapsto F(t,x,\lambda)$ is lsc.

$\displaystyle H_{0}:$ the mapping $\displaystyle\lambda\mapsto\xi(\lambda)$ is continuous from $\displaystyle D$ into $\displaystyle H$.

Assume that $\displaystyle\lambda\mapsto(u(\lambda),h(\lambda))$ is a continuous map from $\displaystyle D$ into $\displaystyle C(T,H)\times L^{2}(T,H)$. We can find a continuous map $\displaystyle p:D\rightarrow L^{2}(T)$ such that

[TABLE]

(see hypothesis $\displaystyle H(F)^{\prime}_{2}(iii)$).

In what follows, by $\displaystyle e(h,\lambda)\in W_{p}(0,b)$ we denote the unique solution of the Cauchy problem

[TABLE]

with $\displaystyle h\in L^{2}(T,H)$.

We have the following approximation result.

Proposition 8**.**

Assume that hypotheses $\displaystyle H(A)_{2},H(F)^{\prime}_{2},H_{0}$ hold, $\displaystyle\lambda\mapsto(u(\lambda),h(\lambda))$ is a continuous map from $\displaystyle D$ into $\displaystyle C(T,H)\times L^{2}(T,H)$ with $\displaystyle u(\lambda)=e(h(\lambda),\lambda)$, $\displaystyle\epsilon>0$ and $\displaystyle p:D\rightarrow L^{2}(T)_{+}$ is a continuous map such that

[TABLE]

Then there exists a continuous map $\displaystyle\lambda\mapsto(x(\lambda),f(\lambda))$ from $\displaystyle D$ into $\displaystyle C(T,H)\times L^{2}(T,H)$ such that

[TABLE]

and $\displaystyle|x(\lambda)(t)-u(\lambda)(t)|\leqslant b\epsilon e^{\tau(t)}+\int^{t}_{0}p(\lambda)(s)e^{\tau(t)-\tau(s)}ds$ for all $\displaystyle t\in T$ with $\displaystyle\tau(t)=\int^{t}_{0}k(s)ds$.

Proof.

Consider the multifunction $\displaystyle R_{\epsilon}:D\rightarrow 2^{L^{1}(T,H)}$ defined by

[TABLE]

This multifunction has nonempty, decomposable values and it is lsc (see Hu and Papageorgiou [22, Lemma 8.3, p. 239]). Hence $\displaystyle\lambda\mapsto\overline{R_{\epsilon}(\lambda)}$ has the same properties. So, we can find a continuous map $\displaystyle\gamma_{0}:D\rightarrow L^{1}(T,H)$ such that

[TABLE]

Let $\displaystyle x_{1}(\lambda)\in W_{p}(0,b)$ be the unique solution of the following Cauchy problem

[TABLE]

Then as in the proof of Proposition 7, we can generate by induction two sequences

[TABLE]

satisfying properties $\displaystyle(a)\rightarrow(d)$ listed in the proof of Proposition 7.

As before (see the proof of Proposition 7), we have

[TABLE]

From this inequality and property $\displaystyle(d)$ of the sequences (see the proof of Proposition 7), we infer that

[TABLE]

are both Cauchy uniformly in $\displaystyle\lambda\in K\subseteq D$ compact (recall that $\displaystyle\lambda\mapsto p(\lambda)$ is continuous, hence locally bounded). So, we have

[TABLE]

and both maps $\displaystyle D\ni\lambda\mapsto\hat{x}(\lambda)\in C(T,H)$ and $\displaystyle D\ni\lambda\mapsto\hat{\gamma}(\lambda)\in L^{1}(T,H)$ are continuous. Moreover, we have $\displaystyle\hat{\gamma}(\lambda)\in S^{2}_{F(\cdot,\hat{x}(\lambda)(\cdot),\lambda)}$ (see the proof of Theorem 6) and that $\displaystyle\lambda\mapsto\hat{\gamma}(\lambda)$ is continuous from $\displaystyle D$ into $\displaystyle L^{2}(T,H)$. If $\displaystyle x(\lambda)=e(\hat{\gamma}(\lambda),\lambda)$, then

[TABLE]

From the triangle inequality, we have

[TABLE]

Using property $\displaystyle(d)$ (see the proof of Proposition 7), we have

[TABLE]

So, finally we can write that

[TABLE]

∎

We want to strengthen Proposition 7, and require that the selection $\displaystyle\vartheta(\cdot)$ passes through a preassigned solution. We mention that an analogous result for differential inclusions in $\displaystyle\mathbb{R}^{N}$ with $\displaystyle A\equiv 0$, was proved by Cellina and Staicu [10].

We start with a simple technical lemma.

Lemma 9**.**

If $\displaystyle\{u_{k}\}_{k=0}^{N}\subseteq L^{1}(T,H)$ and $\displaystyle\{T_{k}(\xi)\}^{N}_{k=0}$ is a partition of $\displaystyle T=[0,b]$ with endpoints which depend continuously on $\displaystyle\xi\in H$, then there exists $\displaystyle\hat{d}\in L^{1}(T)_{+}$ for which the following holds:

“given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta>0$ such that

$\displaystyle|\xi-\xi^{\prime}|\leqslant\delta\Rightarrow\left|\overset{N}{\underset{\mathrm{k=0}}{\sum}}\chi_{T_{k}(\xi)}(t)u_{k}(t)-\overset{N}{\underset{\mathrm{k=0}}{\sum}}\chi_{T_{k}(\xi^{\prime})}(t)u_{k}(t)\right|\leqslant\hat{d}(t)\chi_{C}(t)$

with $\displaystyle C\subseteq T$ measurable and $\displaystyle|C|_{1}\leqslant\epsilon$”.

Proof.

We have

[TABLE]

We set $\displaystyle\hat{d}(t)=\overset{N}{\underset{\mathrm{k=0}}{\sum}}|u_{k}(t)|\in L^{1}(T)_{+}$. From the hypothesis concerning the partition $\displaystyle\{T_{k}(\xi)\}^{N}_{k=0}$ of $\displaystyle T$, we see that given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta>0$ such that

[TABLE]

with $\displaystyle C\subseteq T$ measurable, $\displaystyle|C|_{1}\leqslant\epsilon$. Then

[TABLE]

The proof is now complete. ∎

With this lemma, we can produce a continuous selection of the solution multifunction $\displaystyle\xi\mapsto S(\xi)$, which passes through a preassigned point.

Proposition 10**.**

If hypotheses $\displaystyle H(A)_{2},H(F)_{2}$ hold, $\displaystyle K\subseteq H$ is compact, $\displaystyle\xi_{0}\in K$ and $\displaystyle v\in S(\xi_{0})$, then there exists a continuous map $\displaystyle\psi:K\rightarrow C(T,H)$ such that

[TABLE]

Proof.

Since $\displaystyle v\in S(\xi_{0})$, we have

[TABLE]

with $\displaystyle f\in S^{2}_{F(\cdot,v(\cdot))}$. Given $\displaystyle g\in L^{2}(T,H)$, we consider the unique solution of the Cauchy problem

[TABLE]

In what follows, by $\displaystyle e(g,\xi)\in W_{p}(0,b)$ we denote the unique solution of problem (68) and we set $\displaystyle\mu_{0}(\xi)=e(f,\xi)$. An easy application of the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [22, Theorem 2.14, p. 158]), gives $\displaystyle\gamma_{0}(\xi)\in L^{2}(T,H)$ such that

[TABLE]

Let $\displaystyle\vartheta>0$ we define

[TABLE]

The family $\displaystyle\{B_{\delta(\xi)}(\xi)\}_{\xi\in K}$ is an open cover of the compact set $\displaystyle K$. So, we can find $\displaystyle\{\xi_{k}\}^{N}_{k=0}\subseteq K$ such that $\displaystyle\{B_{\delta(\xi_{k})}(\xi_{k})\}^{N}_{k=0}$ is a finite subcover of $\displaystyle K$. Let $\displaystyle\{\eta_{k}\}^{N}_{k=0}$ be a locally Lipschitz partition of unity subordinated to the finite subcover. We define

[TABLE]

The endpoints in these intervals are continuous functions of $\displaystyle\xi$. We consider the following Cauchy problem

[TABLE]

Problem (69) has a unique solution $\displaystyle\mu_{1}(\xi)\in W_{p}(0,b)$. Let

[TABLE]

Using Lemma 9, we can find $\displaystyle\hat{d}\in L^{1}(T)_{+}$ such that, for any given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta>0$ for which we have

[TABLE]

with $\displaystyle C\subseteq T$ measurable, $\displaystyle|C|_{1}\leqslant\epsilon$. We have $\displaystyle\mu_{1}(\xi^{\prime})=e(\lambda_{0}(\xi^{\prime}),\xi^{\prime})$. As before, exploiting the monotonicity of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)_{2}(ii)$) and using Lemma A.5, p. 157, of Brezis [6], we have

[TABLE]

Let $\displaystyle\epsilon>0$ be given. By the absolute continuity of the Lebesgue integral, we can find $\displaystyle\delta_{1}>0$ such that

[TABLE]

Also, using (70), we can find $\displaystyle\delta\in(0,\epsilon/2)$ such that

[TABLE]

with $\displaystyle C_{1}\subseteq T$ measurable, $\displaystyle|C_{1}|_{1}\leqslant\delta_{1}$. So, returning to (71) and using (72) and (73), we see that

[TABLE]

Therefore $\displaystyle\xi\mapsto\mu_{1}(\xi)$ is continuous from $\displaystyle H$ into $\displaystyle C(T,H)$. Again, with an application of the Yankov-von Neumann-Aumann selection theorem, we obtain $\displaystyle\gamma_{1}(\xi)\in L^{2}(T,H)$ such that

[TABLE]

As in the proof of Proposition 7, we produce inductively two sequences

[TABLE]

which satisfy the following properties:

• (a)

$\displaystyle\mu_{n}(\xi)=e(\lambda_{n-1}(\xi),\xi)$ with $\displaystyle\lambda_{n-1}(\xi)=\overset{N}{\underset{\mathrm{k=0}}{\sum}}\chi_{T_{k}(\xi)}\gamma_{n-1}(\xi_{k})(t),\gamma_{-1}(\xi)=f$;

• (b)

$\displaystyle\xi\mapsto\mu_{n}(\xi)$ is continuous from $\displaystyle K$ into $\displaystyle C(T,H)$;

• (c)

$\displaystyle|\mu_{n}(\xi)(t)-\mu_{n-1}(\xi)(t)|\leqslant\frac{\vartheta}{2^{n+2}n!}\left(\int^{t}_{0}k(s)ds\right)^{n}$ for all $\displaystyle\xi\in K$;

• (d)

$\displaystyle\gamma_{n}(\xi)(t)\in F(t,\mu_{n}(\xi)(t))$ for almost all $\displaystyle t\in T$ and

[TABLE]

So, as induction hypothesis, suppose that we have produced

[TABLE]

which satisfy properties $\displaystyle(a)\rightarrow(d)$ stated above. We set

[TABLE]

As above (see in the first part of the proof the argument concerning the map $\displaystyle\xi\mapsto\mu_{1}(\xi)$), we can show that $\displaystyle\xi\mapsto\mu_{n+1}(\xi)$ is continuous from $\displaystyle K$ into $\displaystyle C(T,H)$. Also, by the monotonicity of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)_{2}(ii)$ and Lemma A.5, p. 157, of Brezis [6]), we have

[TABLE]

Moreover, a standard measurable selection argument, produces a measurable map

$\displaystyle\gamma_{n+1}(\xi):T\rightarrow H,\ \xi\in K$, such that

[TABLE]

This completes the induction process.

Note that

[TABLE]

Therefore, we can say that

[TABLE]

It follows that $\displaystyle\xi\mapsto\psi(\xi)$ is continuous from $\displaystyle K$ into $\displaystyle C(T,H)$.

Note that $\displaystyle T_{0}(\xi_{0})=T=[0,b]$ and so $\displaystyle\mu_{0}(\xi_{0})=e(f,\xi_{0})=v$ (see (67)). Hence $\displaystyle\psi(\xi_{0})=v$. It remains to show that $\displaystyle\psi$ is a selection of the solution multifunction $\displaystyle\xi\mapsto S(\xi)$. By property $\displaystyle(d)$ and hypothesis $\displaystyle H(F)_{2}(ii)$, we have

[TABLE]

Let $\displaystyle\hat{\mu}(\xi)=e\left(\overset{N}{\underset{\mathrm{k=0}}{\sum}}\int^{t}_{0}\chi_{T_{k}(\xi)}(s)\hat{\gamma}(\xi_{k})(s)ds,\xi\right)$. Because

[TABLE]

we have

[TABLE]

∎

4. Optimal Control Problems

In this section we deal with the sensitivity analysis of the optimal control problem (5).

Let $\displaystyle Q(\xi,\lambda)\subseteq W_{p}(0,b)\times L^{2}(T,Y)$ be the admissible “state-control” pairs. First we investigate the dependence of this set on the initial condition $\displaystyle\xi\in H$ and the parameter $\displaystyle\lambda\in E$. Recall that the control space $\displaystyle Y$ is a separable reflexive Banach space and the parameter space $\displaystyle E$ is a compact metric space. To have a useful result on the dependence of $\displaystyle Q(\xi,\lambda)$ on $\displaystyle(\xi,\lambda)\in H\times E$, we introduce the following conditions on the data of the evolution inclusion in problem (5) (the dynamical constraint of the problem).

$\displaystyle H(A)_{3}:$ $\displaystyle A:T\times X\times E\rightarrow 2^{X^{*}}\backslash\{\emptyset\}$ is a multifunction such that

• (i)

for every $\displaystyle(x,\lambda)\in X\times E,\ t\mapsto A_{\lambda}(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, all $\displaystyle\lambda\in E$, $\displaystyle x\mapsto A_{\lambda}(t,x)$ is maximal monotone;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$, all $\displaystyle\lambda\in E$ and all $\displaystyle h^{*}\in A_{\lambda}(t,x)$, we have

[TABLE]

with $\displaystyle\{a_{\lambda}\}_{\lambda\in E}\subseteq L^{p^{\prime}}(T)$ bounded $\displaystyle\{c_{\lambda}\}_{\lambda\in E}\subseteq(0,+\infty)$ bounded and $\displaystyle 2\leqslant p<\infty$;

• (iv)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in X$, all $\displaystyle\lambda\in E$ and all $\displaystyle h^{*}\in A_{\lambda}(t,x)$, we have

[TABLE]

with $\displaystyle\hat{c}>0,\hat{a}\in L^{1}(T)_{+}$;

• (v)

if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then $\displaystyle\frac{d}{dt}+a_{\lambda_{n}}\xrightarrow{PG}\frac{d}{dt}+a_{\lambda}$ as $\displaystyle n\rightarrow\infty$.

Hypotheses $\displaystyle H(A)_{3}(i)\rightarrow(iv)$ are the same as hypotheses $\displaystyle H(A)_{2}(i)\rightarrow(iv)$ for every map $\displaystyle A_{\lambda}$, $\displaystyle\lambda\in E$. The new condition is hypothesis $\displaystyle H(A)_{3}(v)$, which requires elaboration. In the examples that follow, we present characteristic situations where this hypothesis is satisfied.

Example 11**.**

$\displaystyle(a)$* First, we present a situation which will be used in Section 5.*

So, let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with Lipschitz boundary $\displaystyle\partial\Omega$. Let $\displaystyle X=W^{1,p}_{0}(\Omega)$ $\displaystyle(2\leqslant p<\infty)$, $\displaystyle H=L^{2}(\Omega)$, $\displaystyle X^{*}=W^{-1,p^{\prime}}(\Omega)$. Evidently, $\displaystyle(X,H,X^{*})$ is an evolution triple (see Definition 1), with compact embeddings. We consider a map $\displaystyle a(t,z,\xi)$ satisfying the following conditions:

$\displaystyle H(a):$* $\displaystyle a:T\times\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is a map such that*

• (i)

$\displaystyle|a(t,z,0)|\leqslant c_{0}$* for almost all $\displaystyle(t,z)\in T\times\Omega$;*

• (ii)

for every $\displaystyle\xi\in\mathbb{R}^{N},(t,z)\mapsto a(t,z,\xi)$ is measurable;

• (iii)

$\displaystyle|a(t,z,\xi_{1})-a(t,z,\xi_{2})|\leqslant\hat{c}(1+|\xi_{1}|+|\xi_{2}|)^{p-1-\alpha}|\xi_{1}-\xi_{2}|^{\alpha}$* for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle\xi_{1},\xi_{2}\in\mathbb{R}^{N}$, with $\displaystyle\hat{c}_{1}>0$, $\displaystyle\alpha\in\left(0,1\right]$;*

• (iv)

$\displaystyle(a(t,z,\xi_{1})-a(t,z,\xi_{2}),\xi_{1}-\xi_{2})_{\mathbb{R}^{N}}\geqslant\hat{c}_{2}|\xi_{1}-\xi_{2}|^{p}$* for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle\xi_{1},\xi_{2}\in\mathbb{R}^{N}$, $\displaystyle\xi_{1}\neq\xi_{2}$, with $\displaystyle\hat{c}_{2}>0$.*

We consider the operator $\displaystyle A:T\times X\rightarrow X^{*}$ defined by

[TABLE]

Using the nonlinear Green’s identity (see Gasinski and Papageorgiou [20, p. 210]), we have

[TABLE]

with $\displaystyle B(t,x)(\cdot)=a(t,\cdot,Dx(\cdot))\in L^{p^{\prime}}(\Omega,\mathbb{R}^{N})$ for all $\displaystyle(t,x)\in T\times X$.

Now consider a sequence $\displaystyle\{a_{n}(t,z,\xi)\}_{n\geqslant 1}$ of such maps satisfying

[TABLE]

with $\displaystyle\vartheta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ being an increasing function which is continuous at $\displaystyle r=0$ and $\displaystyle\vartheta(0)=0$. We assume that for almost all $\displaystyle t\in T$, $\displaystyle a_{n}(t,\cdot,\cdot)\stackrel{{\scriptstyle G}}{{\rightarrow}}a(t,\cdot,\cdot)$ in the sense of Defranceschi [12]. By Svanstedt [40] we have

[TABLE]

$\displaystyle(b)$* We can allow multivalued maps, provided that we drop the $\displaystyle t$-dependence. So, we consider multivalued maps $\displaystyle a(z,\xi)$ which satisfy the following conditions:*

$\displaystyle H(a)^{\prime}:$* $\displaystyle a:\Omega\times\mathbb{R}^{N}\rightarrow 2^{\mathbb{R}^{N}}\backslash\{\emptyset\}$ is a measurable map such that*

• (i)

$\displaystyle a(\cdot,\cdot)$* is measurable;*

• (ii)

for almost all $\displaystyle z\in\Omega,\ \xi\mapsto a(z,\xi)$ is maximal monotone;

• (iii)

for almost all $\displaystyle z\in\Omega$, all $\displaystyle\xi\in\mathbb{R}^{N}$ and all $\displaystyle y\in a(z,\xi)$, we have

[TABLE]

We again consider the evolution triple

[TABLE]

and consider the multivalued map $\displaystyle A:X\rightarrow 2^{X^{*}}\backslash\{\emptyset\}$ defined by

[TABLE]

We consider a sequence $\displaystyle\{a_{n}(z,\xi)\}_{n\geqslant 1}$ of such maps and assume that $\displaystyle a_{n}\stackrel{{\scriptstyle G}}{{\rightarrow}}a$ in the sense of Defranceschi [12]. Then by Denkowski, Migorski and Papageorgiou [14] we have

[TABLE]

$\displaystyle(c)$* A third situation leading to hypothesis $\displaystyle H(A)_{3}(v)$ is the following one. We consider maps $\displaystyle A_{\lambda}(t,x)$ satisfying the following conditions:*

$\displaystyle H(A)^{\prime}_{3}:$* $\displaystyle A:T\times X\times E\rightarrow X^{*}$ is a map such that*

• (i)

$\displaystyle||A_{\lambda}(t+\tau,x)-A_{\lambda}(t,x)||\leqslant O(\tau)(1+||x||^{p-1})$* for all $\displaystyle t,t+\tau\in T$, all $\displaystyle x\in X$, all $\displaystyle\lambda\in E$;*

• (ii)

for all $\displaystyle(t,\lambda)\in T\times E,\ x\mapsto A_{\lambda}(t,x)$ is hemicontinous;

• (iii)

for all $\displaystyle t\in T$, all $\displaystyle x,u\in X$, all $\displaystyle\lambda\in E$, we have

[TABLE]

• (iv)

if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then for all $\displaystyle t\in T$, $\displaystyle A_{\lambda_{n}}(t,\cdot)\stackrel{{\scriptstyle G}}{{\rightarrow}}A_{\lambda}(t,\cdot)$ (this means that for all $\displaystyle x^{*}\in X^{*}$, $\displaystyle A^{-1}_{\lambda_{n}}(t,x^{*})\stackrel{{\scriptstyle w}}{{\rightarrow}}A^{-1}_{\lambda}(t,x)$, see Denkowski, Migorski and Papageorgiou **[16, Definition 3.8.20, p. 478]**).

Under these conditions, by Kolpakov [26], we have

[TABLE]

Next, we introduce the conditions on the multifunctions $\displaystyle F$ and $\displaystyle G$ involved in the dynamics of (5).

$\displaystyle H(F)_{3}:$ $\displaystyle F:T\times H\times E\rightarrow P_{f_{c}}(H)$ is a multifunction such that

• (i)

for all $\displaystyle(x,\lambda)\in H\times E$, $\displaystyle t\mapsto F(t,x,\lambda)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, all $\displaystyle x,y\in H$ and all $\displaystyle\lambda\in E$, we have

[TABLE]

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$ and all $\displaystyle\lambda\in E$, we have

[TABLE]

with $\displaystyle\{a_{\lambda}\}_{\lambda\in E}\subseteq L^{2}(T)$ and $\displaystyle\{c_{\lambda}\}_{\lambda\in E}\subseteq(0,+\infty)$ bounded;

• (iv)

for almost all $\displaystyle t\in T$, all $\displaystyle x\in H$ and all $\displaystyle\lambda,\lambda^{\prime}\in E$, we have

[TABLE]

with $\displaystyle\beta(r)\rightarrow 0^{+}$ as $\displaystyle r\rightarrow 0^{+}$ and $\displaystyle w(t,\cdot)$ bounded on bounded sets.

$\displaystyle H(G):$ $\displaystyle G:T\times Y\times E\rightarrow P_{f_{c}}(H)$ is a multifunction such that

• (i)

for all $\displaystyle(u,\lambda)\in Y\times E,\ t\mapsto G(t,u,\lambda)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, all $\displaystyle\lambda\in E$, $\displaystyle u\mapsto G(t,u,\lambda)$ is concave (that is, $\displaystyle{\rm Gr}\,G(t,\cdot,\lambda)\subseteq Y\times H$ is concave, see Hu and Papageorgiou [22], Definition 1.1 and Remark 1.2, p. 585) and $\displaystyle(u,\lambda)\mapsto G(t,u,\lambda)$ is $\displaystyle h$-continuous;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle u\in U(t,\lambda)$, all $\displaystyle\lambda\in E$

[TABLE]

with $\displaystyle\{\hat{a}_{\lambda}\}_{\lambda\in E}\subseteq L^{2}(T)$ bounded.

Remark 5**.**

A typical situation resulting to a concave multifunction $\displaystyle u\mapsto G(t,u,\lambda)$, is when

[TABLE]

with $\displaystyle B_{\lambda}(t)\in\mathcal{L}(Y,H)$ and $\displaystyle C(t,\lambda)\in P_{f_{c}}(H)$ for all $\displaystyle(t,\lambda)\in T\times E$.

Another situation, leading to the concavity of $\displaystyle G(t,\cdot,\lambda)$, is when $\displaystyle H$ is an ordered Hilbert space and $\displaystyle g_{\lambda},\tilde{g}_{\lambda}:T\times Y\rightarrow H$ are two Carathéodory maps such that for almost all $\displaystyle t\in T$

[TABLE]

We set $\displaystyle G(t,u,\lambda)=\{h\in H:g_{\lambda}(t,u)\leqslant h\leqslant\tilde{g}_{\lambda}(t,u)\}$. Then $\displaystyle G(t,\cdot,\lambda)$ is concave.

Finally we impose conditions on the control constraint $\displaystyle U(t,\lambda)$.

$\displaystyle H(U):$ $\displaystyle U:T\times E\rightarrow P_{f_{c}}(Y)$ is a multifunction such that

• (i)

for all $\displaystyle\lambda\in E,t\mapsto U(t,\lambda)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T,\ \lambda\mapsto U(t,\lambda)$ is $\displaystyle h$-continuous;

• (iii)

$\displaystyle|U(t,\lambda)|\leqslant\tilde{a}_{\lambda}(t)$ for almost all $\displaystyle t\in T$, all $\displaystyle\lambda\in E$, with $\displaystyle\{\tilde{a}_{\lambda}\}_{\lambda\in E}\subseteq L^{2}(T)$ bounded.

Proposition 12**.**

If hypotheses $\displaystyle H(A)_{3},H(F)_{3},H(G),H(U)$ hold and $\displaystyle(\xi_{n},\lambda_{n})\rightarrow(\xi,\lambda)$ in $\displaystyle H\times E$, then

[TABLE]

Proof.

Let $\displaystyle(x,u)\in K_{seq}(s\times w)-\limsup\limits_{n\rightarrow\infty}Q(\xi_{n},\lambda_{n})$. By definition (see Section 2), we can find a subsequence $\displaystyle\{m\}$ of $\displaystyle\{n\}$ and $\displaystyle(x_{m},u_{m})\in Q(\xi_{m},\lambda_{m})$, $\displaystyle m\in\mathbb{N}$ such that

[TABLE]

For every $\displaystyle m\in\mathbb{N}$, we have

[TABLE]

with $\displaystyle f_{m},g_{m}\in L^{2}(T,H)$ such that

[TABLE]

We deduce by hypotheses $\displaystyle H(F)_{3}(iii),H(G)(iii)$ and Theorem 6 and its proof that

[TABLE]

So, from (76) we obtain

[TABLE]

By (78) and hypotheses $\displaystyle H(F)_{3}(iii),H(G)(iii)$ it is clear that

[TABLE]

Hence, we may assume (at least for a subsequence), that

[TABLE]

Proposition 6.6.33 on p. 521 of Papageorgiou and Kyritsi [34], implies that

[TABLE]

Fix $\displaystyle t\in T\backslash N$ and let $\displaystyle y\in w-\limsup\limits_{m\rightarrow\infty}F(t,x_{m}(t),\lambda_{m})$. By definition, we know that there exists a subsequence $\displaystyle\{k\}$ of $\displaystyle\{m\}$ and $\displaystyle y_{k}\in F(t,x_{k}(t),\lambda_{k})$ for all $\displaystyle k\in\mathbb{N}$ such that $\displaystyle y_{k}\stackrel{{\scriptstyle w}}{{\rightarrow}}y$ in $\displaystyle H$ as $\displaystyle k\rightarrow\infty$. The function $\displaystyle v\mapsto d(v,F(t,x(t),\lambda))$ is continuous and convex, hence weakly lower semicontinuous. Therefore

[TABLE]

On the other hand, we have

[TABLE]

Using hypotheses $\displaystyle H(F)_{3}(ii),(iv)$, we have

[TABLE]

Then we obtain from (82) and (83) that

[TABLE]

Next, for each $\displaystyle m\in\mathbb{N}$, we have

[TABLE]

Let $\displaystyle h\in L^{2}(T,H)$ and let $\displaystyle(\cdot,\cdot)_{L^{2}(T,H)}$ denote the inner product of $\displaystyle L^{2}(T,H)$ (recall that $\displaystyle L^{2}(T,H)^{*}=L^{2}(T,H)$). Then

[TABLE]

The concavity of $\displaystyle G(t,\cdot,\lambda)$ (see hypothesis $\displaystyle H(G)(ii)$), implies that the function $\displaystyle u\mapsto\sigma(h(t),G(t,u,\lambda))$ is concave. Since $\displaystyle E$ is a complete metric space, it can be isometrically embedded, by the Arens-Eells theorem (see Gasinski and Papageorgiou [21, Theorem 4.143, p. 655]), as a closed subset of a separable Banach space (recall that $\displaystyle E$ is compact). So, by Balder [3], we have

[TABLE]

Since $\displaystyle h\in L^{2}(T,H)$ is arbitrary, it follows that

[TABLE]

Let $\displaystyle y_{m}\in W_{p}(0,b)$ be the unique solution of the Cauchy problem

[TABLE]

Hypothesis $\displaystyle H(A)_{3}(v)$ implies that

[TABLE]

with $\displaystyle y\in W_{p}(0,b)$ being the unique solution of the Cauchy problem

[TABLE]

(see Section 2). From (77) and (85) and the monotonicity of $\displaystyle A_{\lambda_{m}}(t,\cdot)$ (see hypothesis $\displaystyle H(A)_{3}(ii)$), we have

[TABLE]

Recalling that

[TABLE]

it follows from (87) that

[TABLE]

Next, we will prove the second convergence of the proposition.

So, let $\displaystyle(x,u)\in Q(\xi,\lambda)$. By definition we have

[TABLE]

with $\displaystyle g\in L^{2}(T,H)$ satisfying

[TABLE]

For every $\displaystyle v\in L^{2}(T,Y)$, we have

[TABLE]

Hypothesis $\displaystyle H(U)(ii)$ and the dominated convergence theorem imply that

[TABLE]

Hence Proposition 6.6.22 on p. 518 of Papageorgiou and Kyritsi [34] implies that we can find $\displaystyle u_{n}\in S^{2}_{U(\cdot,\lambda_{n})}$ ($\displaystyle n\in\mathbb{N}$) such that

[TABLE]

Then hypothesis $\displaystyle H(G)(ii)$ guarantees that we can find

[TABLE]

such that

[TABLE]

Given $\displaystyle\xi^{\prime}\in H$, let $\displaystyle S(\xi^{\prime})\subseteq W_{p}(0,b)$ be the set of solutions of the Cauchy problem

[TABLE]

Let $\displaystyle K=\{\xi_{n},\xi\}_{n\geqslant 1}\subseteq H$. This is a compact set in $\displaystyle H$. Invoking Proposition 10 (with $\displaystyle\xi_{0}=\xi$), we produce a continuous map $\displaystyle\psi:K\rightarrow C(T,H)$ such that

[TABLE]

Let $\displaystyle y_{n}=\psi(\xi_{n})$ ($\displaystyle n\in\mathbb{N}$) and use Proposition 8 to find $\displaystyle x_{n}\in W_{p}(0,b)$ solution of the Cauchy problem

[TABLE]

for which we have

[TABLE]

with $\displaystyle\epsilon>0$, $\displaystyle\tau(t)=\int^{t}_{0}k(s)ds,\ \eta_{n}\in L^{1}(T),\ \eta_{n}\rightarrow 0$ in $\displaystyle L^{1}(T)$.

So, we obtain

[TABLE]

Since $\displaystyle\epsilon>0$ is arbitrary, it follows that

[TABLE]

Finally, we have

[TABLE]

Since $\displaystyle(x_{n},u_{n})\in Q(\xi_{n},\lambda_{n})$ ($\displaystyle n\in\mathbb{N}$) and $\displaystyle u_{n}\rightarrow u$ in $\displaystyle L^{2}(T,Y)$, we conclude that

[TABLE]

∎

An immediate consequence of the above proposition is the following corollary concerning the multifunction $\displaystyle(\xi,\lambda)\mapsto Q(\xi,\lambda)$ of admissible state-control pairs.

Corollary 13**.**

If hypotheses $\displaystyle H(A)_{3},H(F)_{3},H(G),H(U)$ hold, then the multifunction $\displaystyle Q:H\times E\rightarrow 2^{C(T,H)\times L^{2}(T,Y)}\backslash\{\emptyset\}$ is lsc and sequentially closed in $\displaystyle C(T,H)\times L^{2}(T,Y)_{w}$ (that is, $\displaystyle{\rm Gr}\,Q\subseteq H\times E\times C(T,H)\times L^{2}(T,Y)_{w}$ is sequentially closed).

Now we bring the cost functional into the picture. The hypotheses on the integrands $\displaystyle L(t,x,\lambda)$ and $\displaystyle H(t,u,\lambda)$ are the following.

$\displaystyle H(L):$ $\displaystyle L:T\times H\times E\rightarrow\mathbb{R}$ is an integrand such that

• (i)

for every $\displaystyle(x,\lambda)\in H\times E$, $\displaystyle t\mapsto L(t,x,\lambda)$ is measurable;

• (ii)

if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then for all $\displaystyle x\in H$ we have $\displaystyle L(\cdot,x,\lambda_{n})\stackrel{{\scriptstyle w}}{{\rightarrow}}L(\cdot,x,\lambda)$ in $\displaystyle L^{1}(T)$;

• (iii)

for almost all $\displaystyle t\in T$, all $\displaystyle x,y\in H$ and all $\displaystyle\lambda\in E$, we have

[TABLE]

where $\displaystyle|x|\vee|y|=\max\{|x|,|y|\}$ and $\displaystyle\rho(t,r)$ is a Carathéodory function on $\displaystyle T\times\mathbb{R}_{+}$ with values in $\displaystyle(0,+\infty)$ such that

[TABLE]

with $\displaystyle\beta_{\vartheta}\in L^{1}(T)_{+},\vartheta>0$.

$\displaystyle H(H):$ $\displaystyle H:T\times Y\times E\rightarrow\mathbb{R}$ is an integrand such that

• (i)

for all $\displaystyle(u,\lambda)\in Y\times E,\ t\mapsto H(t,u,\lambda)$ is measurable;

• (ii)

for almost all $\displaystyle t\in T$, all $\displaystyle v\in E$, $\displaystyle u\mapsto H(t,u,\lambda)$ is convex and for almost all $\displaystyle t\in T$, all $\displaystyle u\in Y$, $\displaystyle\lambda\mapsto H(t,u,\lambda)$ is continuous;

• (iii)

for almost all $\displaystyle t\in T$ and all $\displaystyle(u,\lambda)\in Y\times E$, we have

[TABLE]

with $\displaystyle a\in L^{\infty}(T)$.

$\displaystyle H(\hat{\psi}):$ $\displaystyle\hat{\psi}:H\times E\rightarrow\mathbb{R}$ is a continuous function.

Using the direct method of the calculus of variations, we can produce optimal admissible state-control pairs for problem (5).

Proposition 14**.**

If hypotheses $\displaystyle H(A)_{3},H(F)_{3},H(G),H(U),H(L),H(H)$ and $\displaystyle H(\hat{\psi})$ hold, then for every $\displaystyle(\xi,\lambda)\in H\times E$ we can find $\displaystyle(x^{*},u^{*})\in Q(\xi,\lambda)$ such that

[TABLE]

Proof.

Let $\displaystyle\{(x_{n},u_{n})\}_{n\geqslant 1}\subseteq Q(\xi,\lambda)$ be a minimizing sequence for problem (5). So, we have

[TABLE]

Theorem 6 and hypothesis $\displaystyle H(U)$ imply that

[TABLE]

is relatively $\displaystyle w\times w$-compact (respectively, $\displaystyle s\times w$-compact). So, by the Eberlein-Smulian theorem and by passing to a suitable subsequence if necessary, we can say that

[TABLE]

Then (92) and Proposition 12 imply that

[TABLE]

Also, (92), hypothesis $\displaystyle H(L)(iii)$ and the dominated convergence theorem, imply that

[TABLE]

In addition, as before (see the proof of Proposition 12), using Theorem 2.1 of Balder [3], obtain

[TABLE]

Finally, (92) and hypothesis $\displaystyle H(\hat{\psi})$ imply that

[TABLE]

We deduce from (93), (94), (95), (96) that

[TABLE]

This concludes the proof. ∎

We are now ready for the main sensitivity results concerning problem (5). The first one establishes the Hadamard well-posedness of the problem.

Theorem 15**.**

If hypotheses $\displaystyle H(A)_{3},\ H(F)_{3},\ H(G),\ H(L),\ H(H)$ and $\displaystyle H(\hat{\psi})$ hold, then the value function $\displaystyle m:H\times E\rightarrow\mathbb{R}$ of problem (5) is continuous.

Proof.

Let $\displaystyle(\xi_{n},\lambda_{n})\rightarrow(\xi,\lambda)$ in $\displaystyle H\times E$. Let $\displaystyle(x,u)\in Q(\xi,\lambda)$ such that

[TABLE]

Invoking Proposition 12, we can find $\displaystyle(x_{n},u_{n})\in Q(\xi_{n},\lambda_{n})$ for all $\displaystyle n\in\mathbb{N}$ such that

[TABLE]

We claim that

[TABLE]

To this end note that

[TABLE]

First, we estimate the first summand in the right-hand side of (4). Using hypothesis $\displaystyle H(L)(iii)$, we have

[TABLE]

Let $\displaystyle M=\sup\limits_{n\geqslant 1}||x_{n}||_{C(T,H)}<+\infty$ (see (97)). Then

[TABLE]

Next, we estimate the second term on the right-hand side of (4).

Let $\displaystyle\vartheta>2||x||_{C(T,H)}$ and let $\displaystyle\beta_{\vartheta}\in L^{1}(T)_{+}$ as postulated by hypothesis $\displaystyle H(L)(iii)$. Given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta>0$ such that

“if $\displaystyle C\subseteq T$ is measurable with $\displaystyle|C|_{1}\leqslant\delta$,

[TABLE]

Here, we use the absolute continuity of the Lebesgue integral. Invoking the Scorza-Dragoni theorem (see Papageorgiou and Kyritsi [34, Theorem 6.2.9, p. 471]), we can find $\displaystyle T_{1}\subseteq T$ closed with $\displaystyle|T\backslash T_{1}|\leqslant\frac{\delta}{2}$ and $\displaystyle\rho|_{T_{1}\times\mathbb{R}_{+}}$ is continuous. Since $\displaystyle\rho(t,0)=0$, we can find $\displaystyle\delta_{1}>0$ such that

[TABLE]

Recall that simple functions are dense in $\displaystyle L^{p}(T,H)$. Using this fact, the property that $\displaystyle L^{p}(T,H)$-convergence implies pointwise convergence for almost all $\displaystyle t\in T$ for at least a subsequence and invoking Egorov’s theorem, we can find $\displaystyle T_{2}\subseteq T$ closed and $\displaystyle s:T\rightarrow H$ a simple function such that

[TABLE]

We set $\displaystyle T_{3}=T_{1}\cap T_{2}$. This is a closed subset of $\displaystyle T$ with $\displaystyle|T\backslash T_{3}|_{1}\leqslant\delta$. We have

[TABLE]

Similarly, we show that

[TABLE]

Let $\displaystyle s(t)=\overset{N}{\underset{\mathrm{k=1}}{\sum}}v_{k}\chi_{C_{k}}(t)$ with $\displaystyle v_{k}\in H,C_{k}\subseteq T$ measurable. Using hypothesis $\displaystyle H(L)(ii)$, we can find $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

It follows from (104), (105), (106) that

[TABLE]

This convergence and (100) imply that (98) (our claim) is true.

Next, we consider the integral functional

[TABLE]

For every $\displaystyle\lambda\in E,u\mapsto\Phi(u,\lambda)$ is convex (see hypothesis $\displaystyle H(H)(ii)$). Also, hypothesis $\displaystyle H(H)$ (iii) implies that in a neighborhood of every $\displaystyle u\in L^{2}(T,Y)$

[TABLE]

hence

[TABLE]

(see Papageorgiou and Kyritsi [34, Theorem 1.2.3, p. 13]). Therefore it follows that

[TABLE]

Finally, (97) and hypothesis $\displaystyle H(\hat{\psi})$ imply that

[TABLE]

By (98), (107), (108), we have

[TABLE]

From Proposition 14 we know that for every $\displaystyle n\in\mathbb{N}$, we can find $\displaystyle(x_{n},u_{n})\in Q(\xi_{n},\lambda_{n})$ such that

[TABLE]

As in the proof of Theorem 6, we can show that $\displaystyle\{x_{n}\}_{n\geqslant 1}\subseteq W_{p}(0,b)$ is bounded. In addition, hypothesis $\displaystyle H(U)$ implies that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq L^{2}(T,Y)$ is bounded. So, by passing to a suitable subsequence if necessary, we may assume that

[TABLE]

By (111) and (8), we also have

[TABLE]

Then (111), (112) and Proposition 12 imply that

[TABLE]

Moreover, reasoning as in the proof of Theorem 6, we show that

[TABLE]

By (113) and the first part of the proof, we have

[TABLE]

In addition, (111) and hypotheses $\displaystyle H(H)(ii),\ H(\hat{\psi})$ imply

[TABLE]

Therefore, from (110) we see that

[TABLE]

We infer from (109) and (114) that

[TABLE]

∎

For every $\displaystyle(\xi,\lambda)\in H\times E$, we introduce the set $\displaystyle\Sigma(\xi,\lambda)$ of optimal state-control pairs, that is,

[TABLE]

By Proposition 14, we know that for every $\displaystyle(\xi,\lambda)\in H\times E$, $\displaystyle\Sigma(\xi,\lambda)\neq\emptyset$. For this multifunction we can prove the following useful continuity property.

Theorem 16**.**

If hypotheses $\displaystyle H(A)_{3},\ H(F)_{3},\ H(U),\ H(L),\ H(H)$ and $\displaystyle H(\hat{\psi})$ hold, then the multifunction $\displaystyle\Sigma:H\times E\rightarrow 2^{C(T,H)\times L^{2}(T,Y)}\backslash\{\emptyset\}$ is sequentially usc into $\displaystyle C(T,H)\times L^{2}(T,Y)_{w}$.

Proof.

Let $\displaystyle C\subseteq C(T,H)\times L^{2}(T,Y)_{w}$ be sequentially closed. We need to show that

[TABLE]

is closed in $\displaystyle H\times E$ (see Section 2). To this end, let $\displaystyle\{(\xi_{n},\lambda_{n})\}_{n\geqslant 1}\subseteq\Sigma^{-}(C)$ and assume that

[TABLE]

Let $\displaystyle(x_{n},u_{n})\in\Sigma(\xi_{n},\lambda_{n})\cap C$, $\displaystyle n\in\mathbb{N}$. We know from the proof of Theorem 15 that at least for a subsequence, we have

[TABLE]

By (115) and Proposition 12, we have

[TABLE]

Also, we know from the proof of Theorem 15 that

[TABLE]

Moreover, from (115) and since $\displaystyle C\subseteq C(T,H)\times L^{2}(T,Y)_{w}$ is sequentially closed, we deduce that $\displaystyle(x,u)\in\Sigma(\xi,\lambda)\cap C$. Therefore $\displaystyle\Sigma^{-}(C)\subseteq H\times E$ is closed and this proves the desired sequential upper semicontinuity of the multifunction $\displaystyle(\xi,\lambda)\mapsto\Sigma(\xi,\lambda)$. ∎

5. Application to Distributed Parameter Systems

In this section we present an application to a class of multivalued parabolic optimal control problems.

So, let $\displaystyle T=[0,b]$ and let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a Lipschitz boundary $\displaystyle\partial\Omega$. We examine the following nonlinear, multivalued parabolic optimal control problem:

[TABLE]

Here, $\displaystyle a_{\lambda}:\Omega\times\mathbb{R}^{N}\rightarrow 2^{\mathbb{R}^{N}}$ ($\displaystyle\lambda\in E$) is a family of multifunctions as in Example 11(b). For the other data of problem (123), we introduce the following conditions:

$\displaystyle H(F_{1}):$ $\displaystyle F_{1}:T\times\Omega\times\mathbb{R}\times E\rightarrow P_{f_{c}}(\mathbb{R})$ is a multifunction such that

• (i)

for all $\displaystyle(x,\lambda)\in\mathbb{R}\times E,(t,z)\mapsto F_{1}(t,z,x,\lambda)$ is measurable;

• (ii)

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle x,y\in\mathbb{R}$, all $\displaystyle\lambda\in E$, we have

[TABLE]

with $\displaystyle k_{1}\in L^{1}(T,L^{\infty}(\Omega))$;

• (iii)

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle x\in\mathbb{R}$, all $\displaystyle\lambda\in E$, we have

[TABLE]

with $\displaystyle\hat{a}_{1}\in L^{2}(T\times\Omega),\hat{c}_{1}>0;$

• (iv)

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle x\in\mathbb{R}$, all $\displaystyle\lambda,\lambda^{\prime}\in E$, we have

[TABLE]

with $\displaystyle\beta(r)\rightarrow 0$ as $\displaystyle r\rightarrow 0^{+}$ and $\displaystyle w\in L^{\infty}_{loc}(\Omega\times\mathbb{R}_{+})$.

Remark 6**.**

Consider the multifunction $\displaystyle F(t,z,x,\lambda)$ defined by

[TABLE]

with $\displaystyle f,\hat{f}:T\times\Omega\times\mathbb{R}\times E\rightarrow\mathbb{R}$ two functions such that

• •

for all $\displaystyle(x,\lambda)\in\mathbb{R}\times E$, $\displaystyle(t,z)\mapsto f(t,z,x,\lambda),\hat{f}(t,z,x,\lambda)$ are both measurable;

• •

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle x,x^{\prime}\in\mathbb{R}$, all $\displaystyle\lambda,\lambda^{\prime}\in E$, we have

[TABLE]

with $\displaystyle k,\hat{k}\in L^{1}(T,L^{\infty}(\Omega))$.

Then this multifunction satisfies hypotheses $\displaystyle H(F_{1})$.

$\displaystyle H(g):$ $\displaystyle g:T\times\Omega\times E\rightarrow\mathbb{R}$ is a Carathéodory function (that is, for all $\displaystyle\lambda\in E$, $\displaystyle(t,z)\rightarrow g(t,z,\lambda)$ is measurable and for almost all $\displaystyle(t,z)\in T\times\Omega$, $\displaystyle\lambda\rightarrow g(t,z,\lambda)$ is continuous) and for almost all $\displaystyle(t,z)\in T\times\Omega$ and all $\displaystyle\lambda\in E$, we have $\displaystyle|g(t,z,\lambda)|\leqslant M$ with $\displaystyle M>0$.

$\displaystyle H(r):$ $\displaystyle r:T\times E\rightarrow\mathbb{R}_{+}$ is a Carathéodory function (that is, for all $\displaystyle\lambda\in E$, $\displaystyle t\mapsto r(t,\lambda)$ is measurable and for almost all $\displaystyle t\in T$, $\displaystyle\lambda\rightarrow r(t,\lambda)$ is continuous) and for almost all $\displaystyle t\in T$, all $\displaystyle\lambda\in E$, we have

[TABLE]

with $\displaystyle a\in L^{2}(T).$

Now, we introduce the conditions on the two integrands involved in the cost functional problem (123).

$\displaystyle H(L_{1}):$ $\displaystyle L:T\times\Omega\times\mathbb{R}\times E\rightarrow\mathbb{R}$ is an integrand such that

• (i)

for all $\displaystyle(x,\lambda)\in\mathbb{R}\times E,(t,z)\mapsto L_{1}(t,z,x,\lambda)$ is measurable;

• (ii)

if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then for all $\displaystyle x\in L^{2}(\Omega)$ we have $\displaystyle L_{1}(\cdot,\cdot,x(\cdot),\lambda_{n})\stackrel{{\scriptstyle w}}{{\rightarrow}}L_{1}(\cdot,\cdot,x(\cdot),\lambda)$ in $\displaystyle L^{1}(T\times\Omega)$;

• (iii)

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle x,y\in\mathbb{R}$, all $\displaystyle\lambda\in E$

[TABLE]

with $\displaystyle\rho(t,z,r)$ Carathéodory, $\displaystyle\rho(t,z,0)=0$ for almost all $\displaystyle(t,z)\in T\times\Omega$ and for almost all $\displaystyle(t,z)$, all $\displaystyle r\in[0,\vartheta]$ we have

[TABLE]

with $\displaystyle\beta_{\vartheta}\in L^{1}(T\times\Omega)$.

$\displaystyle H(H)_{1}:$ $\displaystyle H_{1}:T\times\Omega\times\mathbb{R}\times E\rightarrow\mathbb{R}$ is an integrand such that

• (i)

for all $\displaystyle(x,\lambda)\in\mathbb{R}\times E$, $\displaystyle(t,z)\mapsto H_{1}(t,z,x,\lambda)$ is measurable;

• (ii)

for almost all $\displaystyle(t,z)\in T\times\Omega$, $\displaystyle u\mapsto H_{1}(t,z,u,\lambda)$ is convex for all $\displaystyle\lambda\in E$, while $\displaystyle\lambda\mapsto H_{1}(t,z,u,\lambda)$ is continuous for all $\displaystyle u\in\mathbb{R}$;

• (iii)

for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle|u|\leqslant r_{\lambda}(t,z)$, all $\displaystyle\lambda\in E$, we have

[TABLE]

with $\displaystyle\{\hat{a}_{\lambda}\}_{\lambda\in E}\subseteq L^{2}(T\times\Omega)$ bounded.

We consider the following evolution triple:

[TABLE]

Since $\displaystyle 2\leqslant p<\infty$, the Sobolev embedding theorem implies that in this triple the embeddings are compact.

For every $\displaystyle\lambda\in E$, let $\displaystyle A_{\lambda}:X\rightarrow 2^{X^{*}}\backslash\{\emptyset\}$ be the multivalued map defined by

[TABLE]

This map is maximal monotone and if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then

[TABLE]

(see Example 11(b)). So, hypotheses $\displaystyle H(A)_{3}$ hold. In fact, we can have $\displaystyle t$-dependence at the expense of assuming that $\displaystyle a_{\lambda}$ is single-valued. So, we assume that $\displaystyle a_{\lambda}(t,z,\xi)$ satisfies the conditions of Example 11 (a). Then the map $\displaystyle A_{\lambda}:T\times X\rightarrow X^{*}$ is defined by

[TABLE]

In fact, by the nonlinear Green’s identity (see Gasinski and Papageorgiou [20, p. 210]), we have

[TABLE]

As we have already mentioned in Example 11(a), we know from Svanstedt [40] that if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then

[TABLE]

and so hypotheses $\displaystyle H(A)_{3}$ hold.

As a special case of interest, we consider the situation where the elliptic differential operator is a weighted $\displaystyle p$-Laplacian, that is,

[TABLE]

Here, for every $\displaystyle\lambda\in E,\ a_{\lambda}:T\times\Omega\rightarrow\mathbb{R}$ is a measurable function such that

• •

$\displaystyle 0<\hat{c}_{1}\leqslant a_{\lambda}(t,z)\leqslant\hat{c}_{2}$ for almost all $\displaystyle(t,z)\in T\times\Omega$, all $\displaystyle\lambda\in E$;

• •

if $\displaystyle\lambda_{n}\rightarrow\lambda$ in $\displaystyle E$, then for almost all $\displaystyle t\in T$,

[TABLE]

For this case we consider the following parametric (with parameter $\displaystyle\lambda\in E$) family of convex (in $\displaystyle\xi\in\mathbb{R}^{N}$) integrands:

[TABLE]

Then the convex conjugate of $\displaystyle\varphi_{\lambda}(t,z,\cdot)$ is given by

[TABLE]

By hypothesis we have that

[TABLE]

We introduce the integral functional $\displaystyle\Phi_{\lambda}$ defined by

[TABLE]

By Marcellini and Sbordone [29], we know that (124) implies

[TABLE]

with $\displaystyle\Gamma_{seq}(w)$ denoting the sequential $\displaystyle\Gamma$-convergence of $\displaystyle\Phi_{\lambda_{n}}(t,\cdot)$ on $\displaystyle W^{1,p}_{0}(\Omega)_{w}$ (see Buttazzo [7]). Then it follows from Defranceschi [12, Theorem 3.3] that

[TABLE]

and so we conclude from Svanstedt [40] that

[TABLE]

Also, let, $\displaystyle Y=H=L^{2}(\Omega)$ and

[TABLE]

Then hypotheses $\displaystyle H(F_{1}),\ H(g),\ H(r)$ imply that conditions $\displaystyle H(F)_{3},\ H(G),\ H(U)$ hold. So, the dynamics of (123) are described by an evolution inclusion similar to the one in problem (5).

Finally let

[TABLE]

Hypotheses $\displaystyle H(L_{1}),\ H(H_{1})$ imply that conditions $\displaystyle H(L),\ H(H)$ respectively hold.

So, we can apply Theorems 15 and 16 and obtain the following result concerning the variational stability of problem (123).

Proposition 17**.**

If the maps $\displaystyle a_{\lambda}$ are as above and hypotheses $\displaystyle H(F_{1}),\ H(g),\ H(r),\ H(L_{1})$, $\displaystyle H(H_{1})$ hold, then for every $\displaystyle(\xi,\lambda)\in L^{2}(\Omega)\times E$, problem (123) admits optimal pairs (that is, $\displaystyle\Sigma(\xi,\lambda)\neq\emptyset$) and

[TABLE]

Acknowledgements. This research was supported in part by the SRA grants P1-0292-0101, J1-6721-0101 and J1-7025-0101.