\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2002(2002), No. 28, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2002 Southwest Texas State University.} \vspace{1cm}} \begin{document} \title[\hfilneg EJDE--2001/28\hfil Optimal control] {Optimal control for a nonlinear age-structured population dynamics model} \author[B. Ainseba, S. Ani\c{t}a, \& M. Langlais\hfil EJDE--2001/28\hfilneg] {Bedr'Eddine Ainseba, Sebastian Ani\c{t}a, \& Michel Langlais} \address{Bedr'Eddine Ainseba \hfill\break Math\'{e}matiques Appliqu\'{e}es de Bordeaux, UMR CNRS 5466\\ Case 26, UFR Sciences et Mod\'elisation\\ Universit\'{e} Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France} \email{ainseba@sm.u-bordeaux2.fr} \address{Sebastian Ani\c{t}a \hfill\break Faculty of Mathematics, University ``Al.I. Cuza'' and\\ Institute of Mathematics, Romanian Academy \\ Ia\c{s}i 6600, Romania} \email{sanita@uaic.ro} \address{Michel Langlais \hfill\break Math\'{e}matiques Appliqu\'{e}es de Bordeaux, UMR CNRS 5466\\ Case 26, UFR Sciences et Mod\'elisation\\ Universit\'{e} Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France} \email{langlais@sm.u-bordeaux2.fr} \date{} \thanks{Submitted January 4, 2003. Published March 16, 2003.} \subjclass[2000]{35D10, 49J20, 49K20, 92D25} \keywords{Optimal control, optimality conditions, \hfill\break\indent age-structured population dynamics} \begin{abstract} We investigate the optimal harvesting problem for a nonlinear age-dependent and spatially structured population dynamics model where the birth process is described by a nonlocal and nonlinear boundary condition. We establish an existence and uniqueness result and prove the existence of an optimal control. We also establish necessary optimality conditions. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction and setting of the problem} We consider a general mathematical model describing the dynamics of a single species population with age dependence and spatial structure. Let $u(x,t,a) $ be the distribution of individuals of age $a\ge 0$ at time $t\ge 0$ and location $x$ in $\overline{\Omega }$. Here $\Omega $ is a bounded open subset of $\mathbb{R}^{N}$, $N\in \{ 1,2,3\} $, with a suitably smooth boundary $\partial \Omega $. Thus \begin{equation} P(x,t)=\int_{0}^{A_{\dagger }}u(x,t,a)~da \label{e1} \end{equation} is the total population at time $t$ and location $x$, where $A_{\dagger }$ is the maximal age of an individual. Let $\beta (x,t,a,P(x,t))\ge 0$ be the natural fertility-rate, and let $\mu (x,t,a,P(x,t))\ge 0$ be the natural death-rate of individuals of age $a$ at time $t$ and location $x$. We also assume that the flux of population takes the form $k\nabla u(x,t,a)$ with $k>0$, where $\nabla $ is the gradient vector with respect to the spatial variable $x$. In this paper we are concerned with the optimal harvesting problem on the time interval $(0,T)$, $T>0$, subject to an external supply of individuals $f(x,t,a)\geq 0$ and to a specific harvesting effort $v(x,t,a)$, where $(x,t,a) \in Q=\Omega \times (0,T) \times (0,A_{\dagger })$. So, we deal with the problem of finding the harvesting effort $v$ in order to obtain the best harvest; i.e., Maximize, over all $v\in \mathcal{V}$, the value of \begin{equation} \int_{Q}v(x,t,a) g(x,t,a) u^{v}(x,t,a) \,dx\,dt\ da \,, \label{P} \end{equation} where $g$ is a given bounded function, and $u^{v}$ is the solution of \begin{equation} \begin{array}{ll} \partial _{t}u+\partial _{a}u-k\Delta _{x}u+\mu (x,t,a,P(x,t)) u=f-v u, & \mbox{\rm in } Q \\ \displaystyle {\frac{\partial u}{\partial \eta }}(x,t,a) =0, & \mbox{\rm on } \Sigma \\ \displaystyle u(x,t,0)=\int_{0}^{A_{\dagger }}\beta (x,t,a,P(x,t))u(x,t,a)\,da, & \mbox{\rm in } \Omega \times (0,T) \\ u(x,0,a)=u_{0}(x,a), & \mbox{\rm in } \Omega \times (0,A_{\dagger }), \end{array}\label{e2} \end{equation} where $\Sigma =\partial \Omega \times (0,T) \times (0,A_{\dagger }) $. From a biological point of view $g(x,t,a)\geq 0$ is a weight (the price of an individual of age $a$ at time $t$ and location $x$) and $u_0(x,a)\geq 0$ is the initial distribution of population. The set of controllers is $$ \mathcal{V}=\Big\{ v\in L^{2}(Q) : \zeta _{1}(x,t,a) \leq v(x,t,a) \leq \zeta _{2}(x,t,a) \ \ \mbox{a.e. } (x,t,a) \in Q \Big\} $$ for some $\zeta _{1}, \zeta _{2}\in L^{\infty }(Q)$, $0\leq \zeta _{1}(x,t,a)\leq \zeta _{2}(x,t,a)$ a.e. in $Q$. The harvesting problem for linear initial value age-structured population has been previously studied in Brokate \cite{b1,b2}, Gurtin et al \cite{g1,g2}, Murphy et al \cite{m1} and the periodic case in Ani\c{t}a et al \cite{a1}. We assume the following hypotheses: \begin{itemize} \item[(H1)] The fertility rate satisfies $\beta \in L^{\infty }(Q\times \mathbb{R})$, $\beta (x,t,a,P) \geq 0$ a.e. $(x,t,a,P) \in Q\times \mathbb{R}$ and is decreasing and locally Lipschitz continuous with respect to the variable $P$ \item[(H2)] The mortality rate satisfies $\mu \in L_{\rm loc}^{\infty }(\overline{\Omega }\times [ 0,T]\times [ 0,A_{\dagger }) \times \mathbb{R})$, and $\mu$ is increasing and locally Lipschitz continuous with respect to the variable $P$, $\mu (x,t,a,P) \geq \mu _{0}(a,t) \geq 0$ a.e. $(x,t,a,P) \in Q\times \mathbb{R}$, where $\mu _{0}\in L_{\rm loc}^{\infty }([ 0,T] \times [ 0,A_{\dagger }))$ and $$\int^{A_{\dagger }}\mu _{0}(t+a-A_{\dagger},a)\,da=+\infty , \quad \mbox{a.e. } t\in (0,T) \,. $$ \end{itemize} The last condition in (H2) implies that each individual in the population dies before age $A_{\dagger }$. In addition, we assume the following on $u_{0}$, $f$, $g$: \begin{itemize} \item[(H3)] $u_{0}\in L^{\infty }(\Omega \times (0,A_{\dagger}))$, $u_{0}(x,a) \geq 0$ a.e. $(x,a) \in \Omega \times (0,A_{\dagger })$. \item[(H4)] $f,\; g\in L^{\infty }(Q)$, $f(x,t,a) ,\; g(x,t,a) \geq 0$ a.e. $(x,t,a) \in Q$. \end{itemize} This paper is organized as follows. In Section 2 we prove that under the assumptions listed above and for any $v\in \mathcal{V}$, \eqref{e2} admits a unique and nonnegative solution. A compactness result for the same system is also proved. In Section 3 we treat the existence of an optimal control for problem \eqref{P}. Section 4 is devoted to the deduction of the necessary optimality conditions for the optimal harvesting problem. \section{Existence, uniqueness and compactness of solutions} The first part of this section is devoted to the existence and uniqueness of solutions to system \eqref{e2}, under assumptions (H1)--(H4) and with $v\in \mathcal{V}$ fixed. By a solution to \eqref{e2}, we mean a function $u\in L^2(Q)$ which belongs to $C(\overline{S};L^2(\Omega ))\cap AC(S;L^2(\Omega ))\cap L^2(S;H^1(\Omega ))\cap L^2_{loc}(S;L^2(\Omega ))$, for almost any characteristic line $S$ of equation $a-t=const., \ (t,a)\in (0,T)\times (0,A_{\dagger })$ and satisfies \begin{equation*} \begin{array}{ll} Du(x,t,a)-k\Delta _x u(x,t,a)+\mu (x,t,a,P(x,t))u(x,t,a) & \\ =f(x,t,a)-v(x,t,a)u(x,t,a), & \mbox{a.e. in } Q \\ \displaystyle {\frac{\partial u}{\partial \eta }}(x,t,a) =0, & \mbox{a.e. in } \Sigma \\ \displaystyle \lim_{h\to 0^+}u(x,t+h,h) =\int_0^{A_{\dagger }}\beta (x,t,a,P(x,t))u(x,t,a)\,da, & \mbox{a.e. in } \Omega \times (0,T) \\ \displaystyle \lim_{h\to 0^+}u(x,h,a+h)=u_0(x,a), & \mbox{a.e. in } \Omega \times (0,A_{\dagger }), \end{array} \end{equation*} where $P$ is given by \eqref{e1} and $Du$ denotes the directional derivative $$ Du(x,t,a)=\lim_{h\to 0}{\frac{1}{h}}\big[ u(x,t+h,a+h)-u(x,t,a)\big]. $$ \begin{theorem} \label{thm1} For any $v\in \mathcal{V}$, \eqref{e2} admits a unique and nonnegative solution $u^{v}$ which belongs to $L^{\infty }(Q)$. \end{theorem} \paragraph{Proof} Denote by $\Lambda $ the mapping $\Lambda :\widetilde{u}\mapsto u^{\widetilde{u},{v}}$, where $u^{\widetilde{u},{v}}$ is the solution of \begin{equation*} \begin{array}{ll} Du-k\Delta _{x}u+\mu (x,t,a,\widetilde{P}(x,t)) u=f-v(x,t,a)u, &\quad (x,t,a)\in Q \\ \displaystyle \frac{\partial u}{\partial \eta }(x,t,a) =0, &\quad (x,t,a)\in \Sigma \\ u(x,t,0)=\displaystyle \int_{0}^{A_{\dagger }}\beta (x,t,a,\widetilde{P}(x,t) )u(x,t,a)\,da, &\quad (x,t)\in \Omega \times (0,T) \\ u(x,0,a)=u_{0}(x,a), &\quad (x,a)\in \Omega \times (0,A_{\dagger }) , \end{array} \end{equation*} with $\widetilde{P}(x,t) = \int_{0}^{A_{\dagger }} \widetilde{u}(x,t,a)\,da$. Let $L^p_+(Q)=\{ u\in L^p(Q): u(x,t,a)\geq 0 \mbox{ a.e. in } Q \}$. Then the mapping $\Lambda $ is well defined from $L_{+}^2(Q)$ to $L_{+}^2(Q)$; see Garroni et al \cite{g3}. The comparison result in Garroni et al \cite{g3} and in Langlais \cite{l1} implies $$ 0\leq u^{\widetilde{u},{v}}(x,t,a)\leq \overline{u}(x,t,a) \quad \mbox{a.e. in }Q, $$ where $\overline{u}\in L_+^{\infty }(Q)$ is the solution of \eqref{e2} corresponding to a null mortality rate and to a maximal fertility rate equal to $\| \beta \| _{L^{\infty }( Q\times \mathbb{R})}$. For any $\widetilde{u}_1, \; \widetilde{u}_2\in L^2(Q)$ we denote $ \widetilde{P}_i(x,t)= \int_{0}^{A_{\dagger }} \widetilde{u}_i(x,t,a)\,da$, with $(x,t)\in \Omega \times (0,T)$, and $i\in\{1,2\}$. Using now the definition of $\Lambda $ we obtain \begin{align*} \int_{Q_t}[D(\Lambda \widetilde{u}_1-\Lambda \widetilde{u}_2)- k\Delta _x(\Lambda \widetilde{u}_1-\Lambda \widetilde{u}_2)+ \mu(x,s,a,\widetilde{P}_1(x,t)) (\Lambda \widetilde{u}_1-\Lambda \widetilde{u}_2)&\\ +(\mu (x,s,a,\widetilde{P}_1)-\mu (x,s,a,\widetilde{P}_2)) \widetilde{u}_2+v (\Lambda \widetilde{u}_1-\Lambda \widetilde{u}_2)] (\Lambda \widetilde{u}_1-\Lambda \widetilde{u}_2)\,dx\,ds\,da &= 0, \end{align*} where $Q_t=\Omega \times (0,t)\times (0,A_{\dagger})$, $t\in (0,T)$. Using Gauss-Ostrogradski's formula and the Lipschitz continuity of $\mu $ and $\beta $ with respect to $P$, we get after some calculations that $$ \| (\Lambda \widetilde{u}_{1}-\Lambda \widetilde{u}_{2}) (t) \| _{L^{2}(\Omega \times (0,A_{\dagger }) )}^{2}\leq C\int_{0}^{t}\| (\widetilde{u}_{1}-\widetilde{u} _{2})(s)\| _{L^{2}(\Omega \times (0,A_{\dagger }) )}^{2}\ ds, $$ %250 where $C$ is a positive constant. Banach's fixed point theorem allows us to conclude the existence of a unique fixed point for $\Lambda $. Since the solution $u^v$ satisfies $$0\leq u^{v}(x,t,a)\leq \overline{u}(x,t,a) \quad \mbox{a.e. in }Q $$ and $\overline{u}\in L_+^{\infty}(Q)$, we complete the proof. \hfill$\diamondsuit$ \smallskip For $v\in \mathcal{V}$, let $$ P^{v}(x,t)=\int_{0}^{A_{\dagger }}u^{v}(x,t,a)\,da \quad (x,t)\in \Omega \times(0,T) \,. $$ We shall prove now a compactness result which is one of the main ingredients in the next section. \begin{lemma} \label{lm2} The set $\{ P^{v}; \ v\in \mathcal{V}\} $ is relatively compact in $L^{2}(\Omega \times (0,T))$. \end{lemma} \paragraph{Proof} Because $u^{v}$ is a solution of \eqref{e2}, for any $\varepsilon >0$ small enough we have that $$ P^{v,\varepsilon }(x,t)=\int_{0}^{A_{\dagger }- \varepsilon }u^{v}(x,t,a)\,da, \quad (x,t) \in \Omega \times (0,T) $$ is a solution of \begin{gather*} \begin{aligned} P_{t}^{v,\varepsilon }-k\Delta _{x}P^{v,\varepsilon } &= \int_{0}^{A_{\dagger}-\varepsilon }(f-(\mu (x,t,a,P^{v}(x,t)) +v) u^{v}) da -u^{v}(x,t,A_{\dagger }-\varepsilon)\\ &\quad+\int_{0}^{A_{\dagger }}\beta (x,t,a,P^{v}(x,t) )u^{v}(x,t,a)\,da, \quad \mbox{a.e. in } \Omega \times (0,T) \end{aligned}\\ {\frac {\partial P^{v,\varepsilon }}{\partial \eta }} (x,t) =0, \quad \mbox{a.e. }\partial \Omega \times (0,T) \\ P^{v,\varepsilon }(x,0)=\int_{0}^{A_{\dagger }-\varepsilon } u_{0}(x,a)\,da, \quad \mbox{a.e. in }\Omega \,. \end{gather*} Since $\{vu^{v}\}$ and $\{\mu (\cdot ,\cdot ,\cdot ,P^{v}) u^{v}\}$ are bounded in $L^{\infty }(\Omega \times (0,T)\times (0,A_{\dagger }-\varepsilon ))$, $\{\beta (\cdot ,\cdot ,\cdot,P^{v}) u^{v}\}$ is bounded in $L^{\infty }(\Omega \times (0,T)\times (0,A_{\dagger }))$ and $\{u^{v}(\cdot ,\cdot ,A_{\dagger}-\varepsilon ) \}$ is bounded in $L^{\infty }(\Omega \times (0,T))$ - with respect to $v\in \mathcal{V}$ (as a consequence of the proof of Theorem \ref{thm1}), we conclude that $\{ P_{t}^{v,\varepsilon }-k\Delta_{x}P^{v,\varepsilon }\} $ is bounded in $L^{\infty }(\Omega \times (0,T))$. This implies via Aubin's compactness theorem that for any $\varepsilon >0$ small enough, the set $\{P^{v,\varepsilon };v\in \mathcal{V}\} $ is relatively compact in $L^{2}(\Omega \times (0,T))$. On the other hand $$ | P^{v,\varepsilon }(x,t) -P^{v}(x,t)| \leq \int_{A_{\dagger }-\varepsilon }^{A_{\dagger }}| u^{v}(x,t,a)| \,da \leq \varepsilon \| \overline{u}\|_{L^{\infty }(Q) } \,, $$ for all $\varepsilon >0$, and all $v\in \mathcal{V}$, a.e. $(x,t)$ in $\Omega \times (0,T) $. Combining these two results we conclude the relative compactness of $\{ P^{v};v\in \mathcal{V}\} $ in $L^{2}(Q)$. \hfill$\diamondsuit$ \section{Existence of an optimal control} In this section, we prove the existence of an optimal pair (an optimal control $v^{\ast }$ and the corresponding solution $u^{v^{\ast}}$ for problem \eqref{P}). Indeed we have the following theorem. \begin{theorem} \label{thm3} Problem \eqref{P} admits at least one optimal pair. \end{theorem} \paragraph{Proof} Let $\varphi :\mathcal{V}\to \mathbb{R}^{+}$, be defined by $$ \varphi (v) =\int_{Q}v(x,t,a) g(x,t,a) u^{v}(x,t,a) \,dx\, dt\, da $$ and let $d=\sup_{v\in \mathcal{V}}\varphi (v)$. Since by the proof of Theorem \ref{thm1} $$0\leq \varphi (v) \leq \int_{Q} \zeta _{2}(x,t,a) g(x,t,a) \overline{u}(x,t,a) \,dx\, dt\, da \, , $$ we have $d\in [ 0,+\infty) $. Now let $\{ v_{n}\} _{n\in \mathbb{N}^{*}}\subset \mathcal{V}$ be a sequence such that $$d-\frac{1}{n}< \varphi (v_{n}) \leq d \ . $$ Since $0\leq u^{v_{n}}(x,t,a) \leq \overline{u}(x,t,a)$ a.e. in $Q$, we conclude that there exists a subsequence, also denoted by $\{v_{n}\} _{n\in N^{*}}$, such that $$u^{v_{n}}\to u^{*}\mbox{ \rm weakly in }L^{2}(Q) \ .$$ Using Mazur's theorem we obtain the existence of a sequence $\{\widetilde{u}_{n}\} _{n\in N^{*}}$ such that $$ \widetilde{u}_{n}(x,t,a) =\sum_{i=n+1}^{k_{n}} u^{v_{i}}, \quad \lambda _{i}^{n}\geq 0, \quad \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}=1 $$ and $\widetilde{u}_{n}\to u^{*}\mbox{ \rm in }L^{2}(Q)$. Consider now the sequence of controls $$\widetilde{v}_{n}(x,t,a) = \begin{cases} \frac{\sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}v_{i}(x,t,a) u^{v_{i}}(x,t,a)} { \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}u^{v_{i}}(x,t,a)} & \mbox{if } \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}u^{v_{i}}(x,t,a) \neq 0 \\ \vspace{2mm} \zeta _{1}(x,t,a) , & \mbox{if } \sum_{i=n+1}^{k_{n}}\lambda_{i}^{n}u^{v_{i}}(x,t,a) =0 \,. \end{cases} $$ For these controls we have $\widetilde{v}_n\in \mathcal{V}$. Lemma \ref{lm2} implies the existence of a subsequence, also denoted by $\left\{ v_{n}\right\} _{n\in N^{*}}$ such that \begin{equation} P^{v_{n}}\to P^{*}\quad\mbox{in }L^{2}(\Omega \times (0,T)) \label{e4} \end{equation} and since $u^{v_{n}}\to u^{*}$ weakly in $L^{2}(Q)$, then we obtain that $$ \int_{0}^{A_{\dagger }}u^{v_{n}}(\cdot ,\cdot ,a)\,da\to \int_{0}^{A_{\dagger }}u^{*}(\cdot ,\cdot ,a)\,da\quad \mbox{weakly in } L^{2}(\Omega \times (0,T)). $$ Consequently we get that $$P^{*}(x,t)=\int_{0}^{A_{\dagger }}u^{*}(x,t,a)\,da \quad \mbox{a.e. in }\Omega \times (0,T)\,. $$ We can take a subsequence, also denoted by $\left\{ \widetilde{v}_{n}\right\} _{n\in N^{*}}$, such that \begin{equation*} \widetilde{v}_{n}\to v^{*}\quad \mbox{weakly in }L^{2}(Q), \end{equation*} with $v^{*}\in \mathcal{V}$. It is obvious now that $\widetilde{u}_{n}$ is a solution of \begin{equation} \begin{array}{ll} \displaystyle Du-k\Delta _{x}u+\sum_{i=n+1}^{k_{n}}\lambda _{i}^{n} \mu (x,t,a,P^{v_{i}}(x,t)) u^{v_{i}} & \\ =f-\widetilde{v}_{n}u, & \mbox{\rm in } Q \\ \displaystyle \frac{\partial u}{\partial \eta }(x,t,a) =0, & \mbox{\rm on } \Sigma \\ \displaystyle u(x,t,0)=\int_{0}^{A_{\dagger }} \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n} \beta (x,t,a,P^{v_{i}}(x,t)) u^{v_{i}}\,da, &\mbox{\rm in } \Omega \times (0,T) \\ u(x,0,a)=u_{0}(x,a), & \mbox{\rm in } \Omega \times (0,A_{\dagger }) \,. \end{array} \label{e6} \end{equation} By \eqref{e4} we deduce the existence of a subsequence (also denoted by $\{ v_n\}$) such that \begin{equation*} \begin{gathered} \mu (\cdot ,\cdot ,\cdot ,P^{v_{n}}) \to \mu (\cdot ,\cdot ,\cdot ,P^{*})\quad \mbox{a.e. in }Q \,, \\ \beta (\cdot ,\cdot ,\cdot ,P^{v_{n}}) \to \beta (\cdot ,\cdot ,\cdot ,P^{*}) \quad\mbox{a.e. in }Q \,. \end{gathered} \end{equation*} Since $\widetilde{u}_{n}\to u^{*}\mbox{ \rm in }L^{2}(Q)$, we have $$ \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}\mu (x,t,a,P^{v_{i}}(x,t)) u^{v_{i}}(x,t,a)\to \mu (x,t,a,P^{*}(x,t)) u^{*}(x,t,a) $$ a.e. in $Q$, and $$ \sum_{i=n+1}^{k_{n}}\lambda _{i}^{n} \beta (x,t,a,P^{v_{i}}(x,t)) u^{v_{i}}(x,t,a) \to \beta (x,t,a,P^{*}(x,t)) u^{*}(x,t,a) $$ a.e. in $Q$. Passing to the limit in \eqref{e6} we obtain that $u^{*}$ is the solution of \eqref{e2} corresponding to $v^{*}$. Moreover we have \begin{align*} &\sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}\int_{Q} v_{i}(x,t,a) g(x,t,a) u^{v_{i}}(x,t,a) \, dx\, dt\, da \\ &=\int_{Q}\widetilde{v}_{n}(x,t,a) g(x,t,a) \widetilde{u}_{n}(x,t,a) \,dx\, dt\,da\\ &=\sum_{i=n+1}^{k_{n}}\lambda _{i}^{n}\varphi (v_{i})\to \varphi (v^{*}) \end{align*} (as $n\to +\infty $). We may infer now that $d=\varphi (v^{*})$. \hfill$\diamondsuit$ \section{Necessary optimality conditions} %sec. 4 Concerning the necessary optimality conditions the following result holds under the assumptions (H1)--(H4). \begin{theorem} \label{thm4} Assume $\beta$ and $\mu$ are $C^1$ with respect to $P$. If $(u^{*},v^{*})$ is an optimal pair for \eqref{P} and if $q$ is the solution of \begin{equation*} \begin{array}{ll} -Dq(x,t,a)-k\Delta _{x}q(x,t,a)+\mu (x,t,a,P^{v^{*}}(x,t)) q(x,t,a) \\ \displaystyle +\int_{0}^{A_{\dagger }}\mu _{P}'(x,t,s,P^{*}(x,t)) u^{*}(x,t,s) q(x,t,s) \ ds \\ -\displaystyle \Big( \beta (x,t,a,P^{*}(x,t) )+ \int_{0}^{A_{\dagger }}\beta_{P}'(x,t,s,P^{*}(x,t)) u^{*}(x,t,s) \ ds\Big) q(x,t,0) \end{array} \end{equation*} \vspace{-5mm} \begin{equation} =-v^{*} (g+q)(x,t,a), \quad (x,t,a)\in Q \label{e8} \end{equation} \vspace{-5mm} \begin{equation*} \begin{array}{ll} \displaystyle \frac{\partial q}{\partial \eta }(x,t,a) =0, &\quad (x,t,a)\in \Sigma \\ q(x,t,A_{\dagger })=0, &\quad (x,t)\in \Omega \times (0,T) \\ q(x,T,a)=0, &\quad (x,a)\in \Omega \times (0,A_{\dagger })\, , \end{array} \end{equation*} then we have $$ v^{*}(x,t,a)=\begin{cases} \zeta _{1}(x,t,a) \quad \mbox{if }(g+q)(x,t,a)<0 \\ \zeta _{2}(x,t,a) \quad \mbox{if }(g+q)(x,t,a)>0 \, . \end{cases} $$ Here $\mu '_P$ and $\beta '_p$ are the derivatives of $\mu $ and $\beta $ with respect to $P$. \end{theorem} \paragraph{Proof} Existence and uniqueness of $q$, a solution of \eqref{e8} follows in the same way as the existence and uniqueness of the solution of \eqref{e2}. Since $(v^{*},u^{*}) $ is an optimal pair for \eqref{P} we get \begin{align*} &\int_{Q}v^{*}(x,t,a) g(x,t,a) u^{v^{*}}(x,t,a) \, dx\,dt\,da \\ &\geq \int_{Q} (v^{*}(x,t,a) +\delta v(x,t,a)) g(x,t,a) u^{v^{*}+\delta v}(x,t,a)\,dx\,dt\,da\, \end{align*} for all $\delta$ positive and small enough, for all $v\in L^{\infty }(Q)$ such that %500 \begin{gather*} v(x,t,a)\leq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _2(x,t,a) \\ v(x,t,a)\geq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _1(x,t,a) \,. \end{gather*} This implies \begin{equation} \begin{array}{ll} \displaystyle \int_{Q} v^{*}(x,t,a) g(x,t,a) \frac{u^{v^{*} +\delta v}(x,t,a) -u^{v^{*}}(x,t,a) }{\delta }\,dx\, dt\, da \\ \displaystyle +\int_{Q} v(x,t,a) g(x,t,a) u^{v^{*}+\delta v}(x,t,a) \,dx\,dt\,da \leq 0 \, . \end{array}\label{e9} \end{equation} Using the definition of solution to \eqref{e2} and the comparison result in Garroni et al \cite{g3}, we can prove that for any $v\in L^{\infty }(Q)$ as above, the following convergence holds $$ u^{v^{*}+\delta v}(x,t,a) \longrightarrow u^{v^{*}} (x,t,a) \ \mbox{\rm in } L^{\infty }(0,T; L^2((0,A_{\dagger })\times \Omega )) $$ as $\delta \longrightarrow 0^+$. Let $$ z^{\delta }(x,t,a)= \frac{u^{v^{*}+\delta v}(x,t,a) -u^{v^{*}}(x,t,a) }{\delta }, \quad (x,t,a)\in Q \,. $$ Then the function $z^{\delta }$ is a solution of $$ Dz^{\delta }-k\Delta _xz^{\delta }+\frac{1}{\delta} \big(\mu( x,t,a,P^{v^{*}+\delta v}( x,t) ) u^{v^{*}+\delta v} -\mu ( x,t,a,P^{v^{*}}( x,t))u^{v^{*}}\big) $$ $$ =-v^{*}z^{\delta }-v(x,t,a)u^{v^{*}+\delta v}, \quad (x,t,a)\in Q $$ $$ \frac{\partial z^{\delta }}{\partial \eta } (x,t,a) =0, \quad (x,t,a)\in \Sigma $$ $$ z^{\delta }(x,t,0)= \int_{0}^{A_{\dagger }} \frac{\beta (x,t,a,P^{v^{*}+\delta v}( x,t)) u^{v^{*}+\delta v} -\beta ( x,t,a,P^{v^{*}}( x,t)) u^{v^{*}}}{\delta }\,da, $$ $$\hfil (x,t)\in \Omega \times (0,T) $$ $$ z^{\delta }(x,0,a)=0, \quad (x,a)\in \Omega \times (0,A_{\dagger }) $$ and using again the definition of solution to \eqref{e2} and the comparison result in Garroni et al \cite{g3}, we can prove that $z^{\delta }\to z$ in $L^{\infty }(Q)$ as $\delta \to 0$, where $z$ is the solution of $$ Dz-k\Delta _{x}z+\mu (x,t,a,P^{v^{*}}(x,t)) z(x,t,a) $$ $$ +\mu _{P}'(x,t,a,P^{^{v^{*}}}(x,t)) u^{v^{*}}(x,t,a) \int_{0}^{A_{\dagger }}z(x,t,s) ds $$ $$ =-v^{*}z-v(x,t,a)u^{v^{*}}, \quad (x,t,a)\in Q$$ $$ \frac{\partial z}{\partial \eta }(x,t,a) =0, \quad (x,t,a)\in \Sigma $$ $$ z(x,t,0)=\int_{0}^{A_{\dagger }}\beta ( x,t,a,P^{v^{*}}(x,t) ) z( x,t,a)\,da $$ $$ +\int_{0}^{A_{\dagger }}\Big( \beta _{P}'(x,t,a,P^{^{v^{*}}}( x,t) ) u^{v^{*}} \int_{0}^{A_{\dagger }}z( x,t,s) ds\Big) \,da, \quad (x,t)\in \Omega \times (0,T) $$ $$ z(x,0,a)=0, \quad (x,a)\in \Omega \times ( 0,A_{\dagger }) \,. $$ Passing to the limit in \eqref{e9}, $\delta \to 0^+$, we conclude that $$ \int_{Q} v^{*}(x,t,a) g(x,t,a) z(x,t,a) \,dx\, dt\, da $$ $$ +\int_{Q} v(x,t,a) g(x,t,a) u^{v^{*}}(x,t,a)\,dx\,dt\,da \leq 0 \,, $$ for all $v\in L^{\infty }(Q)$ such that \begin{gather*} v(x,t,a)\leq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _2(x,t,a) \\ v(x,t,a)\geq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _1(x,t,a) \,. \end{gather*} Multiplying \eqref{e8} by $z$ and integrating over $Q$ we get after some calculation that $$ \int_Q(v^{*}gz) (x,t,a) \,dx\,dt\,da =\int_Q(vu^{v^{*}}q) (x,t,a) \,dx\,dt\,da $$ and consequently $$\int_Qv( x,t,a)u^{v^{*}}( x,t,a) ( g+q)( x,t,a)\,dx\,dt\,da\leq 0 \,, $$ for all $v\in L^{\infty }(Q)$ such that \begin{gather*} v(x,t,a)\leq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _2(x,t,a) \\ v(x,t,a)\geq 0 \quad \mbox{if } v^{\ast }(x,t,a)=\zeta _1(x,t,a) \,. \end{gather*} This implies $u^{v^{*}}(g+q) \in N_{\mathcal{V}}(v^{*})$, where $ N_{\mathcal{V}}(v^{*})$ is the normal cone at $\mathcal{V}$ in $v^{\ast }$ (in $L^2(Q))$. For any $(x,t,a)\in Q$ such that $u^{v^{\ast}}(x,t,a)\ne 0$, we conclude $$ v^{\ast }(x,t,a)=\begin{cases} \zeta _{1}(x,t,a) &\mbox{if }(g+q)(x,t,a)<0 \\ \zeta _{2}(x,t,a) &\mbox{if }(g+q)(x,t,a)>0 \,. \end{cases} $$ On the other hand, for any $(x,t,a)\in Q$ such that $u^{v^{\ast}}(x,t,a)=0$, it is obvious that we can change the value of the optimal control $v^{\ast }$ in $(x,t,a)$ with any arbitrary value belonging to $[\zeta _1(x,t,a), \zeta _2(x,t,a)]$ and the state corresponding to this new control is the same and the value of the cost functional also remains the same. The conclusion of Theorem \ref{thm4} is now obvious. \hfill$\diamondsuit$ \begin{thebibliography}{00} \frenchspacing \bibitem{a1} S. Ani\c{t}a, M. Iannelli, M.-Y. Kim and E.-J. Park; \textit{Optimal harvesting for periodic age-dependent population dynamics}, SIAM J. Appl. Math., \textbf{58} (1998), 1648-1666. \bibitem{b1} M. 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