\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A minmax problem for parabolic systems \hfil EJDE--1999/50} {EJDE--1999/50\hfil Sanjay Chawla \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No.~50, pp. 1--18. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A minmax problem for parabolic systems \\ with competitive interactions \thanks{ {\em 1991 Mathematics Subject Classifications:} 49K35, 49K20, 49K22, 35K57, 45K05. \hfil\break\indent {\em Key words and phrases:} optimal control, game theory, saddle point. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted June 16, 1999. Published December 13, 1999.} } \date{} % \author{Sanjay Chawla} \maketitle \begin{abstract} In this paper we model the evolution and interaction between two competing populations as a system of parabolic partial differential equations. The interaction between the two populations is quantified by the presence of non-local terms in the system of equations. We model the whole system as a two-person zero-sum game where the gains accrued by one population necessarily translate into the others loss. For a suitably chosen objective functional(pay-off) we establish and characterize the saddle point of the game. The controls(strategies) are kernels of the interaction terms. \end{abstract} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newcommand{\kernelc}[1]{\int_Q c(x,y,t,\tau) #1\, dyd\tau} \newcommand{\kerneld}[1]{\int_Q d(x,y,t,\tau) #1\, dyd\,\tau} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}{Proposition}[section] \section{Introduction} In 1914 Lancaster \cite{l} proposed the first analytic model for describing combat between two forces. The Lancaster model, a coupled system of ordinary differential equations, fails to account for the spatial movement of opposing forces on the battlefield. To overcome this obvious shortcoming and include spatial dependence, Protopopescu et al. \cite{ps} introduced a more general model, namely, they replaced the system of ordinary differential equations with a system of {\bf semilinear parabolic equations with competitive interactions.} These systems belong to the class of reaction-diffusion systems which lately have become one of the mainstream fields in pure, applied and numerical PDE's \cite{f}. This competition model can be used to represent other phenomenon, e.g., densities of competing biological populations and concentration of chemical reactants. In this paper we consider competitive systems as a two-person zero-sum game which implies that one player's gains necessarily translate to the other players's losses. Each player controls some of the game parameters which the player can manipulate to navigate the evolution of the system towards a desired state. A quantitative measurement of the players performance is modeled in terms of a minimizing(respectively maximizing) functional which depends upon the state of the system and the controls. We will show that under suitable conditions the game admits a unique saddle point and then we will characterize the saddle point of the game as a solution of an optimality system. The optimality system will consist of the parabolic state equations coupled with two adjoint equations. The controls will be the attrition kernels of the non-local interaction terms. The attrition kernels represent the effect of ``weapons'' in the combat model case. In general, the interactions between the two populations are non-local and the kernels measure the range over which one population can affect the other population. Lenhart et al. \cite{lps} considered the steady state case with the operator $-\Delta $ and Lenhart et al. \cite{lp} have also considered the parabolic case where the controls are the source terms. The outline of this part of the paper is the following. In the next section we give the statement of the problem. The payoff(cost) functional is defined and for given controls, the unique solution of the state system is constructed. The existence of the unique saddle point is established in Section 3. In Section 4 the optimality system is defined and the saddle point is represented as the solution of this optimality system. \section{Statement of the Problem} \par Throughout this part of the paper, $C$ denotes generic constants, unless otherwise indicated. Let $\Omega$ be a bounded domain in ${\bf R^m}$ with ${\partial{\Omega}}~{\in}~C^{1,1},$ let ${T}>0$ and $Q = \Omega\times(0,T)$. For $\Gamma > 0,$ define the control set, \[ C_{\Gamma} = \{ c \in L^{\infty}_{+}(Q\times Q)~| ~\|c\| \leq \Gamma \}. \] where $L^{\infty}_{+}$ is the set of positive $L^{\infty}$ functions. For any $c,d \in C_{\Gamma},$ let the pair $(u,v)=(u(c,d),v(c,d))$ denote the solution of the {\it state system} \begin{eqnarray} L_1u(x,t) &=& f(x,t)-u(x,t)\kernelc{v(y,\tau)} \mbox{ on } Q \nonumber \\ L_2v(x,t) &=& g(x,t)-v(x,t)\kerneld{u(y,\tau)} \label{state} \\ &&u=u^0\,,\ v=v^0 \mbox{ on } \Omega \times \{0\} \nonumber \\ &&u=0\,,\ v=0 \mbox{ on } \Sigma = \partial\Omega\times(0,T) \nonumber \end{eqnarray} where $ u,v \in L^2(0,T,H^{1}_0(\Omega)) =V$, and $$ L_{k}u = u_t - (a_{ij}^ku_{x_i})_{x_j} + b^k_iu_{x_i}+c^ku,\, k=1,2\,. $$ Here we have used the summation convention with respect to repeated indices. The solutions $u, v$ represent the concentration of the two competing populations. The sources $f,g$ are given and the attrition kernels $c,d$ are the controls. The first player controls d with the purpose of maximizing $\cal J $ (the payoff); the second player controls c to minimize $\cal J$ (the cost). Given two target functions, $\tilde{u}$, $\tilde{v}$ $\in$ $L^2$, and $K,$ $L$ $\geq$ $0$ and $M,$ $N$ $>$ $0$, the {\it payoff (cost) functional} $\cal J$ is defined by \begin{eqnarray} {\cal J}(c,d) &=&\frac{1}{2}\int_Q \left\{K[u(c,d) - \tilde{u}]^2 - L[v(c,d) -\tilde{v}]^2\right\}\,dx\,dt \nonumber\\ && + \frac{1}{2}\int_Q \int_Q (Nc^2-Md^2)\,dx\,dt\,dy\,d\tau. \end{eqnarray} The {\it saddle point} $(c^{*}, d^{*})$ (if it exists) is defined as a pair of {\it strategies} $(c^{*},d^{*}) \in C_{\Gamma} \times C_{\Gamma}$, such that \[ {\cal J}(c^{*},d^{*})= \sup_{d \in C_{\Gamma}} {\cal J}(c^{*},d)= \inf_{c \in C_{\Gamma}} {\cal J}(c,d^{*}) \] We make the following assumptions: \begin{eqnarray} &a_{ij}^k, b^k_i, c^k \in L^{\infty}(Q) , k=1,2,\, i,j=1,...,m& \\ \label{elliptic} &\theta \zeta_izeta_i \leq a_{ij}^k\zeta_i\zeta_j \leq \theta^{-1} \zeta_i\zeta_i ,\, \theta > 0 \mbox{ for all \boldmath$\zeta$ }\in {\bf R^m} &\\ \label{init} &u^0, v^0 \in L^{\infty}_{+}(\Omega) &\\ &c^k(x,t) > c_0 > 0\,, k= 1,2\,. & \end{eqnarray} Finally to set up solutions in $V$, we define the following bilinear form on $H^{1}_0(\Omega):$ \begin{equation} a^k(t,\phi,\psi) = \int_{\Omega}a^k_{ij}{\phi}_{x_i}{\psi}_{x_j}\,dx + \int_{\Omega}b^k_i{\phi}_{x_i}\psi\, dx + \int_{\Omega}c^k\phi \psi\,dx \mbox{ for each t } \in (0,T). \end{equation} Then for all $\phi , \psi \in L^2(0,T,H^{1}_0(\Omega))$, solutions $(u,v)$ in $V \times V$ of (\ref{state}) satisfy \begin{eqnarray} \lefteqn{ \int_0^T (\langle u_t,\psi\rangle + a^{1}(t,u,\psi))\,dt }\nonumber\\ & = & \int_0^T \int_{\Omega} (f - u(x,t)\kernelc{v(y,\tau)})\psi\,dx\,dt \label{weaksol} \\ \lefteqn{ \int_0^T (\langle v_t,\phi\rangle + a^2(t,v,\phi))\,dt }\nonumber\\ & = & \int_0^T \int_{\Omega} (f - v(x,t)\kerneld{u(y,\tau)})\phi\,dx\,dt \nonumber \end{eqnarray} where $\langle ,\rangle$ denotes the duality between $H_0^{1}(\Omega)$ and $H^{-1}(\Omega)$. Using an iterative scheme we have the following existence result. \begin{prop} {\bf(Existence of solutions of the state system).} Given $c, d \in$ $C_{\Gamma}$, there exists a solution $(u,v)$ of the {\it state system} (\ref{state}) in $V \times V$ and there exists a constant $C_1 > 0$ such that $0 \leq u \leq C_1,$ $ 0 \leq v \leq C_1.$ \end{prop} {\bf Proof:} Since the bilinear form $a^k(t,\phi,\psi)$ is continuous and coercive for all \\ $\phi$, $\psi$ $\in H^{1}_0(\Omega)$, we obtain, using standard linear theory \cite{evans}, positive solutions $u_0$, $\tilde{v}$ in $V$ to \begin{displaymath} \begin{array}{rcll} L_1u_0 & = & f & \mbox{ in $Q$}, \\ u_0 & = & u^0 & \mbox{at $t =0$}, \\ u_0 & = & 0 & \mbox{on $\Sigma$}, \\ \end{array} \end{displaymath} and \begin{displaymath} \begin{array}{rcll} L_2\tilde{v} & = & g & \mbox{ in $Q$}, \\ \tilde{v} & = & v^0 & \mbox{at $t =0$}, \\ \tilde{v} & = & 0 & \mbox{on $\Sigma$}. \\ \end{array} \end{displaymath} Solution $u_0, \tilde{v}$ will be the supersolutions for the iterates to be constructed. Also, by standard existence theory\cite{evans}, there exists a constant $C_1 > 0 $ such that $ \|u_0\|_\infty, \|\tilde{v}\|_\infty \leq C_1. $ Now let $v_0$ be the solution in V of \begin{displaymath} \begin{array}{rcll} L_2v_0 & = & g - \tilde{v}\kerneld{u_0} & \mbox{ in $Q$}, \\ v_0 & = & v^0 & \mbox{at $t =0$}, \\ v_0 & = & 0 & \mbox{on $\Sigma$}. \\ \end{array} \end{displaymath} Let $(u_{k},v_{k})$ be the solutions of the pair of linear Initial - Boundary value problem \begin{equation} \label{iteration} \begin{array}{ll} L_1u_{k} + \mu u_{k} = f - u_{k-1}\int_Qc{v_{k-1}} +\mu u_{k-1} \\ L_2v_{k} + \mu v_{k} = g - v_{k-1}\int_Qd{u_{k-1}} +\mu v_{k-1}& \mbox{ in } Q, \\ u_{k}=u^0, v_{k}= v^0 & \mbox{ at } t=0, \\ u_{k}= 0 , v_{k} = 0 & \mbox{ on } \Sigma, \end{array} \end{equation} where $\mu$ is a constant which makes the right hand side of the first equation in (\ref{iteration}) an increasing function of $u$ and the right hand side of the second equation an increasing function of $v$ for iterates in the range $0 \leq u_{k}, v_{k} \leq C_1.$ %From assumptions (\ref{elliptic}) - (\ref{init}), a large %constant $C_1$ acts as a 'supersolution' and %\[ 0 \leq u^k \leq C_1, \hspace{5mm} 0 \leq v^k \leq C_1 \] We have monotone convergence of the iterates, $$u_{k} \searrow u, \quad v_{k} \nearrow v \mbox{ pointwise},\quad 0 \leq u_{k} \leq C_1, \quad 0 \leq v_{k} \leq C_1\,, $$ through comparision results\cite{lp}. From {\it a priori} estimates of $u_{k},$ $v_{k}$ from the system we get uniform bounds on $\|u_{k}\|_{V},$ $\|v_{k}\|_{V}$. Thus, $u_{k}$ and $v_{k}$ convege weakly to $u$ and $v$ in $V$. Now we show that $u,$ $v$ solve the state system in the sense of (\ref{weaksol}). The uniform bounds on $u_{k}$ and $v_{k}$ in $V$ combined with the state equation give uniform bounds for $(u_{k})_t$ and $(v_{k})_t$ in $L^2(0,T,H^{-1}(\Omega))$. Using compactness results \cite[Chapter 4, Prop. 4.2]{li2} implies that $u_{k}$, $v_{k}$ converge strongly in $L^2(Q)$. Passing to the limit in the weak formulation of the system (\ref{iteration}) we obtain $u=u(c,d)$ and $v=v(c,d)$ which solve the system (\ref{state}).\hfill $\diamondsuit$ \begin{prop} {\bf(Uniqueness of solutions of the state system.)} {For a fixed pair $(c,d)$ in $[C_{\Gamma}]^2$ and for $c_0$ sufficiently large, the state system (\ref{state}) admits a unique solution.} \end{prop} {\bf Proof.} Suppose $(u,v)$ and $(\bar{u},\bar{v})$ solve the state system for given initial conditions $u^0$ and $v^0$. Using test functions $(u-\bar{u})$, $(v - \bar{v})$ and then subtracting the $(\bar{u},\bar{v})$ system from $(u,v)$ system \begin{eqnarray} \label{ueqn} \int_Q[(u-{\bar u})_t(u-{\bar u}) + a^{1}_{ij}(u-{\bar u})_{x_i}(u-{\bar u})_{x_j} +b^{1}_i(u-{\bar u})_{x_i}(u-{\bar u}) + c^{1}(u-{\bar u})^2 \nonumber \\ + \int_Q[(v-{\bar v})_t(v-{\bar v}) + a^2_{ij}(v-{\bar v})_{x_i}(v-{\bar v})_{x_j} +b^2_i(v-{\bar v})_{x_i}(v-{\bar v}) + c^2(v-{\bar v})^2]= \nonumber \\ -\int_Qu(u-\bar{u})\int_Qcv + \int_Q\bar{u}(u-\bar{u})\int_Qc\bar{v} -\int_Qv(v-\bar{v})\int_Q du + \int_Q\bar{v}(v-\bar{v})\int_Qd\bar{u}. \nonumber \\ \end{eqnarray} We will estimate the various terms in the above equality and show that the resulting relationship can only be satisfied if $u=\bar{u}$ and $v=\bar{v}$. Note that\\ (i)\[ \int_Q[(u-\bar{u})_t(u-\bar{u}) + (v-\bar{v})_t(v-\bar{v})] = \frac{1}{2}\int_{\Omega \times T}[(u-\bar{u})^2+(v-\bar{v})^2],\] \\ (ii)\[ \int_Q a_{ij}^{1}(u-\bar{u})_{x_i}(u-\bar{u})_{x_j} \geq \theta \int_Q[|\nabla(u-\bar{u})|^2, \] (iii)\[ \int_Q a_{ij}^2(v-\bar{v})_{x_i}(v-\bar{v})_{x_j} \geq \theta \int_Q[|\nabla(v-\bar{v})|^2, \] where $\theta$ is the ellipticity constant in (\ref{elliptic}),\\ (iv) \[ \left|\int_Qb^{1}_i(u-{\bar u})_{x_i}(u-\bar{u}) \right| \leq C(\epsilon)\int_Q|\nabla (u-\bar{u})|^2 + C(\frac{1}{\epsilon})\int_Q(u-\bar{u})^2 \] and\\ (v) \[ \left|\int_Qb^2_i(v-{\bar v})_{x_i}(v-\bar{v}) \right| \leq C(\epsilon)\int_Q|\nabla (v-\bar{v})|^2 + C(\frac{1}{\epsilon})\int_Q(v-\bar{v})^2. \] We choose $\epsilon$ such that the $C(\epsilon)$'s in (iv) and (v) equal $\theta$ in (ii) and (iii). \\ Now we estimate the double integrals: \begin{eqnarray} \label{uest} \lefteqn{ -\int_Qu(u-\bar{u})\int_Qcv + \int_Q\bar{u}(u-\bar{u})\int_Qc\bar{v}} \nonumber \\ &=& -\int_Qu(u-\bar{u})\int_Qcv + \int_Q\bar{u}(u-\bar{u})\int_Qcv \nonumber \\ &&-\int_Q\bar{u}(u-\bar{u})\int_Qcv + \int_Q\bar{u}(u-\bar{u})\int_Qc\bar{v} \nonumber \\ &=& -\int_Q(u-\bar{u})^2\int_Qcv - \int_Q\bar{u}(u-\bar{u})\int_Qc(v-\bar{v}) \nonumber \\ & \leq& -\int_Q\bar{u}(u-\bar{u})\int_Qc(v-\bar{v}) \nonumber \\ & \leq& C\{ \int_Q(u-\bar{u})^2 + \int_Q(v-\bar{v})^2\}. \end{eqnarray} By choosing $c_0$ sufficiently large, we absorb the $\int_Q(u-\bar{u})^2$ and $\int_Q(v-\bar{v})^2$ terms on the left hand side in (\ref{ueqn}). Thus we get \[ (c_0-C)\{ \int_Q(u-\bar{u})^2 + \int_Q(v-\bar{v})^2\} \leq 0 \] We conclude that $u = \bar{u}$ and $v=\bar{v}$. \hfill $\diamondsuit$ \section{Existence of the Saddle Point} Sufficient conditions for the objective functional\, ${\cal J}(c,d)$ to admit a saddle point are \cite{et}:\\ $(1)$\, The mapping $c \mapsto {\cal J}(c,d)$ is strictly convex and lower semi-continuous in the weak toplogy of $ L^2(Q \times Q).$ \\ $(2)$\, The mapping $d \mapsto {\cal J}(c,d)$ is strictly concave and upper semi-continuous in the weak topology of $ L^2(Q \times Q).$ \\ % We will prove $(1), and $ $(2)$ follows similarly. For $c,\, \bar{c}$ given in $C_{\Gamma}$ define a new function ${\mit J}:{\bf[0,1] } \to {\bf R} $ as \[ {\mit J}(\nu)= {\cal J}({\nu}c + (1-{\nu})\overline{c},d). \] The strict convexity of the map $c\mapsto {\cal J}(c,d)$ is equivalent to showing ${\mit J}''(\nu) > 0$ for all ${\nu}$ in $[0,1]$. Since ${\cal J}$ is a function of the state variables and the state variables themselves are functions of the controls, we begin by estimating the first and second derivatives of $u$ and $v$ with respected to the control $c$. The derivatives involved are directional derivatives, in the distributional sense. We begin by deriving a useful apriori estimate. %Now the presence of the second derivative of ${\mit J}$ forces us, since %${\cal J}$ explicitly contains $u$,$v$ , to define the first and second %``derivatives'' of $u$ and $v$ wrt $c$. Not only will we do that, we will %identify equations and establish apriori bounds which these first and %second ``derivatives'' satisfy. But first, we need a definition and %an apriori estimate. Consider the Gelfand triple \[ V \subset L^2(Q) \subset V' \] and for any $(c,d)$ in $[C_{\Gamma}]^2$ {\bf define} the operator ${\cal L} : V^2 \rightarrow (V^{'})^2$ by the formula \begin{equation} \label{A} { {\cal L}}\left( \begin{array}{c} \zeta \\ \chi \end{array} \right) = \left( \begin{array}{c} L_1\zeta + u\int_Q c\chi + \zeta\int_Q cv \\ L_2\chi + v\int_Q d\zeta + \chi\int_Q du \end{array} \right). \\ \end{equation} \begin{prop}{\label{est}} For any $\epsilon > 0$, there exists $c_0 (\epsilon)$ such that if \[ c^k(x,t) \geq c_0(\epsilon) > 0, k = 1, 2. \] then the solution to \begin{equation} \label{C} { {\cal L}}\left( \begin{array}{c} \zeta \\ \chi \end{array} \right) = \left( \begin{array}{c} \alpha \\ \beta \end{array} \right), \end{equation} with $\zeta = \chi$ at $t = 0$ and $\zeta = \chi = 0$ on $\Sigma$ , satisfies the estimate, \[ \|\zeta\|_{L^2(Q)} + \|\chi\|_{L^2(Q)} \leq \epsilon(\|\alpha\|_{L^2(Q)} + \|\beta\|_{L^2(Q)}). \] \end{prop} {\bf Proof:} We multiply the first equation in (\ref{C}) by $\zeta$ and the second equation by $\chi$. Integrating over $Q$ and using the coercivity of the parabolic operators $L_1,\, L_2$, we get, \begin{eqnarray} \label{B} \lefteqn{\frac{1}{2}\int_{\Omega \times T} (\zeta^2 + \chi^2) + \theta\int_Q(|\nabla\zeta|^2 + |\nabla\chi|^2) + c_0\int_Q(\zeta^2 + \chi^2) }\hspace{40mm} \nonumber\\ &\leq & -\int_Qu\zeta \int_Qc\chi - \int_Qv\chi\int_Qd\zeta + \int_Q\alpha\zeta + \int_Q\beta\chi \nonumber \\ & &\mbox{}- \int_Qb_i^{1}\zeta_{x_i}\zeta -\int_Qb_i^2\chi_{x_i}\chi. \end{eqnarray} Now we use the $L^{\infty}$ bounds of $u,v,c,d$ and the $\epsilon$ - Cauchy inequality to estimate the right hand side of the above inequality.\\ $(i)\quad -\int_Qu\zeta\int_Qc\chi \leq C\int_Q\zeta \int_Q\chi \leq C\int_Q\zeta^2 + C\int_Q\chi^2.$ \\ $(ii)\quad -\int_Qv\chi \int_Qd\zeta \leq C\int_Q\zeta^2+C\int_Q\chi^2.$\\ $(iii)\quad \int_Q\alpha\zeta \leq \epsilon\int_Q\alpha^2 + C_{\epsilon}\int_Q\zeta^2.$ \\ $(iv)\quad \int_Q\beta\chi \leq \epsilon\int_Q\beta^2 + C_{\epsilon}\int_Q\chi^2.$ \\ $(v)\quad \int_Qb_i^{1}\zeta_{x_i}\zeta \leq \frac{\theta}{2}\int_Q|\nabla\zeta|^2 + C_{\theta}\int_Q\zeta^2.$ \\ $(vi)\quad \int_Qb_i^2\chi_{x_i}\chi \leq \frac{\theta}{2}\int_Q|\nabla\chi|^2 + C_{\theta}\int_Q\chi^2.$ \\ Choosing $c{0} $ sufficiently large and estimating the right hand side of $(\ref{B})$ by the above estimates, we arrive at our conclusion. \hfill $\diamondsuit$ \smallskip We now we prove the existence of first derivatives of $u , v $ with respect to the controls. These derivatives satisfy a system with operator ${{\cal L}}$ from (\ref{A}). % \begin{prop} For $ c_0 $ sufficiently large, the mapping \[ c \mapsto (u(c,d),v(c,d)) \in V^2 \] is differentiable in the sense \begin{equation} \begin{array}{lr} \displaystyle \frac{u(c+\beta\overline{c},d) -u(c,d)}{\beta} \rightarrow \zeta & \mbox{ weakly in } V, \end{array} \end{equation} \begin{equation} \begin{array}{lr} \displaystyle \frac{v(c+\beta\overline{c},d) - v(c,d)}{\beta} \rightarrow \chi & \mbox{ weakly in } V, \end{array} \end{equation} as $\beta \rightarrow 0$, for any $(c,d) \in [C_{\Gamma}]^2$ and $\overline{c} \in L^{\infty}(Q)$ such that $c+\beta\overline{c} \in C_{\Gamma}. $\\ Also $(\zeta,\chi) \stackrel{\rm def}{=}((\zeta(c,d;\overline{c},0), (\chi(c,d;\overline{c},0))$ is the unique solution of \begin{equation} \label{firstsys} { {\cal L}} \left( \begin{array}{c} \zeta\\ \chi \end{array} \right) = - \left( \begin{array}{c} u(c,d)\int_Q\overline{c} v(c,d) \\ 0 \end{array} \right). \end{equation} with $\zeta = \chi = 0$ at $t=0$ and $\zeta = \chi = 0$ on $\Sigma$. \end{prop} {\bf Proof:} Let $(u_{\beta},v_{\beta})$ be the solution of the state system corresponding to the controls $(c+\beta\overline{c},d)$.Then multipling the first equation of the state system by $(u_{\beta}-u)$ and the second equation by $(v_{\beta}-v)$ and then integrating over $Q$, we get \\ %\begin{eqnarray} $ \int_Q(u_{\beta}-u)_t(u_{\beta}-u) + \int_Q[a_{ij}^{1}(u_{\beta}-u)_{x_i}(u_{\beta}-u)_{x_j} + \int_Qb_i^{1}(u_{\beta}-u)_{x_i}(u_{\beta}-u) + \int_Qc^{1}(u_{\beta}-u)^2] $ \begin{equation} \label{firstu} = -\int_Q u_{\beta} (u_{\beta}-u)\int_Q(c+\beta\overline{c})v_{\beta} + \int_Qu(u_{\beta}-u)\int_Qcv\\ \end{equation} \mbox{and} \[ \int_Q(v_{\beta}-v)_t(v_{\beta}-v) + \int_Q a_{ij}^2(v_{\beta}-v)_{x_i}(v_{\beta}-v)_{x_j} + \int_Qb_i^2(v_{\beta}-v)_{x_i}(v_{\beta}-v) \] \[ + \int_Qc^2(x,y,t,\tau)(v_{\beta}-v)^2(y,\tau)\,dy\,d\tau\] \[ = \] \begin{equation} \label{firstv} -\int_Q v_{\beta}(v_{\beta}-v)\int_Qdu_{\beta} + \int_Qv(v_{\beta}-v)\int_Qdu.\\ \end{equation} After standard manipulations on the left hand side of (\ref{firstu}) ( using coercivity of the $a^{1}_{ij}$'s and applying $\epsilon$ - Cauchy inequality to separate $\int_Qb_i^{1}(u_{\beta}-u)_{x_i}(u_{\beta}-u)$ ) \begin{eqnarray*} \frac{\theta}{2}\int_Q|\nabla(u_{\beta}-u)|^2 + c_0\int_Q(u_{\beta}-u)^2 & \leq & -\int_Q u_{\beta} (u_{\beta}-u)\int_Q(c+\beta\overline{c})v_{\beta} \\ & + & \int_Qu(u_{\beta}-u)\int_Qcv. \end{eqnarray*} Adding and subtracting $\int_Q u_{\beta} (u_{\beta}-u)\int_Q(c+\beta\overline{c})v$, we get \begin{eqnarray*} \frac{\theta}{2}\int_Q|\nabla(u_{\beta}-u)|^2 + c_0\int_Q(u_{\beta}-u)^2 & \leq & - \int_Q(u_{\beta} (u_{\beta}-u)\int_Q(c+\beta\overline{c})(v_{\beta}-v) \\ & - & \int_Q(u_{\beta}-u)^2\int_Qcv \\ &- & \int_Qu_{\beta}(u_{\beta}-u)\int_Q\beta\overline{c}v. \end{eqnarray*} Again using the apriori bounds of $u,v,c$ ,\\ $\frac{\theta}{2}\int_Q|\nabla(u_{\beta}-u)|^2 + c_0\int_Q(u_{\beta}-u)^2$ \begin{equation} \label{fuest} \leq C\{\|(u_{\beta}-u)\|_{L^2(Q)}^2 + \|(v_{\beta}-v)\|_{L^2(Q)}^2 \} + C \beta^2 \|\bar{c}\|^2_{L^2(Q)}. \end{equation} Similarly (\ref{firstv}) yields\\ $\frac{\theta}{2}\int_Q|\nabla(v_{\beta}-v)|^2 + c_0\int_Q(v_{\beta}-v)^2$ \begin{equation} \label{fvest} \leq C\{\|(u_{\beta}-u)\|_{L^2(Q)}^2 + \|(v_{\beta}-v)\|_{L^2(Q)}^2 \} \end{equation} Combining (\ref{fuest}) and (\ref{fvest}), using $c_0$ large, and dividing across by $ \beta $, we get \[ \left\|\frac{u_{\beta}-u}{\beta}\right\|_{V} + \left\|\frac{v_{\beta}-v}{\beta}\right\|_{V} \leq C. \] \\ Since bounded sets in $V$ are weakly compact, we arrive at the required weak limits. In the weak formulation the system satisfied by \[ \frac{u_{\beta} - u}{\beta}, \frac{v_{\beta} - v}{\beta}\] is, for test functions $\phi, \psi \in V$ \[ \int_Q\left(\frac{(u_{\beta}-u)}{\beta}\right)_t\phi + \int_Q a_{ij}^{1}\left((\frac{u_{\beta}-u)}{\beta}\right) _{x_i}\phi_{x_j} + \int_Qb_i^{1}\left(\frac{(u_{\beta}-u)}{\beta}\right)_{x_i}\phi \] \[ + \int_Qc^{1}\left(\frac{(u_{\beta}-u)}{\beta}\right)\phi \] \[ = \] \begin{equation} \label{firstub} -\frac{1}{\beta}\int_Q u_{\beta} \phi\int_Q(c+\beta\overline{c})v_{\beta} + \frac{1}{\beta}\int_Qu \phi \int_Qcv \end{equation} and \[ \int_Q\left(\frac{(v_{\beta}-v)}{\beta}\right)_t\psi + \int_Q a_{ij}^2\left((\frac{v_{\beta}-v)}{\beta}\right) _{x_i}\psi_{x_j} + \int_Qb_i^2\left(\frac{(v_{\beta}-v)}{\beta}\right)_{x_i}\psi \] \[ + \int_Qc^2\left(\frac{(v_{\beta}-v)}{\beta}\right)\psi \] \[ = \] \begin{equation} \label{firstvb} -\frac{1}{\beta}\int_Q v_{\beta} \psi\int_Qd u_{\beta} + \frac{1}{\beta}\int_Qu \psi \int_Qdu. \end{equation} Letting $\beta \rightarrow 0$ and noting $u_{\beta} \rightarrow u$ ,$\, v_{\beta} \rightarrow v$ we get \begin{equation} \label{zetachi} { {\cal L}} \left( \begin{array}{c} \zeta\\ \chi \end{array} \right)\left(\begin{array}{c} \phi\\ \psi \end{array} \right) = - \int_Q(\phi,\psi) \left( \begin{array}{c} u(c,d)\int_Q\overline{c} v(c,d) \\ 0 \end{array} \right). \end{equation}\, \hfill $\diamondsuit$ \smallskip We have a similar result for the directional derivative of $u, v$ with respect to the control $ d$. \begin{prop} For $ c_0 $ sufficiently large, the mapping \[ d \mapsto (u(c,d),v(c,d)) \in V^2 \] is differentiable in the sense \begin{equation} \begin{array}{lr} \displaystyle \frac{u(c,d+\beta \bar{d}) -u(c,d)}{\beta} \rightarrow \xi & \mbox{ weakly in } V, \end{array} \end{equation} \begin{equation} \begin{array}{lr} \displaystyle \frac{v(c,d + \beta \bar{d}) - v(c,d)}{\beta} \rightarrow \sigma & \mbox{ weakly in } V, \end{array} \end{equation} as $\beta \rightarrow 0$, for any $(c,d) \in [C_{\Gamma}]^2$ and $\bar{d} \in L^{\infty}(Q)$ such that $d+\beta\bar{d} \in C_{\Gamma}. $\\ Also $(\xi,\sigma) \stackrel{\rm def}{=}((\xi(c,d;0,\bar{d}), (\sigma(c,d;0,\bar{d}))$ is the unique solution of \begin{equation} \label{firstsys1} { {\cal L}} \left( \begin{array}{c} \xi\\ \sigma \end{array} \right) = - \left( \begin{array}{c} 0\\ v(c,d)\int_Q\overline{d} u(c,d) \\ \end{array} \right). \end{equation} with $\xi = \sigma = 0$ at $t=0$ and $\xi = \sigma = 0$ on $\Sigma$. \end{prop} We present next the result for the second dervatives of $u, v $ with respect to the controls. \begin{prop} The mapping \[ c \mapsto (u(c,d),v(c,d)) \in V^2 \] admits second derivatives with respect to $c$ in the sense \begin{equation} \label{secdu} \begin{array}{lr} \displaystyle \frac{\zeta(c+\beta\overline{c},d;\overline{c},0)-\zeta(c,d;\overline{c},0)} {\beta} \rightarrow \tau & \mbox{ weakly in } V, \end{array} \end{equation} \begin{equation} \label{secdv} \begin{array}{lr} \displaystyle \frac{\chi(c+\beta\overline{c},d;\overline{c},0)-\chi(c,d;\overline{c},0)} {\beta} \rightarrow \eta & \mbox{ weakly in } V, \end{array} \end{equation} as $\beta \rightarrow 0$, for any $(c,d) \in [C_{\Gamma}]^2$ and $\overline{c} \in L^{\infty}(Q)$ such that $c+\beta\overline{c} \in C_{\Gamma}. $\\ Also $(\tau,\eta) \stackrel{\rm def}{=}((\tau(c,d;\overline{c},0; \overline{c},0),\eta(c,d;\overline{c},0; \overline{c},0))$ is the unique solution of \begin{equation} \label{secdeqn} { {\cal L}} \left( \begin{array}{c} \tau\\ \eta \end{array} \right) = -2\left( \begin{array}{c} \zeta\int_Qc\chi + u\int_Q\overline{c}\chi + \zeta\int_Q\overline{c}v \\ \chi\int_Qd\zeta \end{array} \right). \end{equation} with $\tau = \eta = 0$ at $t=0$ and $\tau = \eta = 0$ on $\Sigma$. \label{second} \end{prop} {\bf Proof:} We denote by $\zeta_{\beta},\chi_{\beta},\zeta,\chi$ the solutions of system (\ref{firstsys}) corresponding to\\ $(c+\beta\overline{c},d;\overline{c},0)$ and $(c,d;\overline{c},0)$ respectively. Using test functions $(\zeta_{\beta} - \zeta, \chi_{\beta} - \chi)$ we subtract the $(\zeta, \chi)$ system from the $(\zeta_{\beta}, \chi_{\beta})$ system \[ \int_Q[(\zeta_{\beta}-\zeta)_t(\zeta_{\beta}-\zeta) + a^{1}_{ij}(\zeta_{\beta}-\zeta)_{x_i}(\zeta_{\beta}-\zeta)_{x_j} +b^{1}_i(\zeta_{\beta}-\zeta)_{x_i}(\zeta_{\beta}-\zeta) + c^{1}(\zeta_{\beta}-\zeta)^2] \] \[ + \int_Q[(\chi_{\beta}-\chi)_t(\chi_{\beta}-\chi) + a^2_{ij}(\chi_{\beta}-\chi)_{x_i}(\chi_{\beta}-\chi)_{x_j}]\] \[ +\int_Q[b^2_i(\chi_{\beta}-\chi)_{x_i}(\chi_{\beta}-\chi) + c^2(\chi_{\beta}-\chi)^2] \] \begin{equation} \label{sdeqn} = \end{equation} \[ - \int_Q(\zeta_{\beta}-\zeta)\zeta_{\beta}\int_Qc(v_{\beta}-v) - \int_Q(\zeta_{\beta}-\zeta)^2\int_Qcv - \int_Q(\zeta_{\beta}-\zeta)u_{\beta}\int_Qc(\chi_{\beta}-\chi) \] \[ -\int_Q(\zeta_{\beta}-\zeta)(u_{\beta}-u)\int_Qc\chi - \int_Q(\zeta_{\beta}-\zeta)\zeta_{\beta}\int_Q\beta\overline{c}v_{\beta} -\int_Q(\zeta_{\beta}-\zeta)u_{\beta}\int_Q\beta\overline{c}\chi_{\beta} \] \[ - \int_Q(\zeta_{\beta}-\zeta)u_{\beta}\int_Q\overline{c}(v_{\beta}-v) - \int_Q(\zeta_{\beta}-\zeta)(u_{\beta}-u)\int_Q\overline{c}v -\int_Q(\chi_{\beta}-\chi)^2\int_Q du_{\beta} \] \[ - \int_Q(\chi_{\beta}-\chi)v_{\beta}\int_Qd(\zeta_{\beta}-\zeta) - \int_Q(\chi_{\beta}-\chi)(v_{\beta}-v)\int_Qd\zeta -\int_Q(\chi_{\beta} - \chi)\chi\int_Qd(u_{\beta} - u). \] We illustrate the estimates for a term with the kernel; \begin{eqnarray} \lefteqn{ \left|\int_Q(\zeta_{\beta}-\zeta)\zeta_{\beta}\int_Qc(v_{\beta}-v) \right| }\nonumber \\ & = & \left|\int_Q\left((\zeta_{\beta}-\zeta)\zeta_{\beta} \int_Qc(v_{\beta}-v)(y,\tau)\,dy\,d\tau\right)(x,t)\,dx\,dt\right| \label{zeta1} \\ & \leq & C\int_Q(\zeta_{\beta}-\zeta)^2(x,t)\,dx\,dt + C \int_Q\left(\zeta_{\beta} \int_Q(v_{\beta} - v)(y,\tau)\,dy\,d\tau\right)^2\,dx\,dt \nonumber\\ & \leq & C\int_Q(\zeta_{\beta}-\zeta)^2(x,t)\,dx\,dt +C\left(\int_Q\zeta_{\beta}^2\,dx\,dt \right)\left(\int_Q (v_{\beta} - v)^2\,dy\,d\tau \right). \label{zeta2} \end{eqnarray} Notice how the specific form of the non-local term was used to derive (\ref{zeta2}) from (\ref{zeta1}). Other such terms are estimated as below: \[ - \int_Q(\zeta_{\beta}-\zeta)^2\int_Qcv \leq 0. \] \[ \int_Q(\zeta_{\beta}-\zeta)u_{\beta}\int_Qc(\chi_{\beta}-\chi) \leq C\int_Q(\zeta_{\beta}-\zeta)^2 + C\int_Q(\chi_{\beta}-\chi)^2. \] \[ \int_Q(\zeta_{\beta}-\zeta)(u_{\beta}-u)\int_Qc\chi \leq C\int_Q(\zeta_{\beta}-\zeta)^2 + C\int_Q(u_{\beta}-u)^2. \] \[ \int_Q(\zeta_{\beta}-\zeta)\zeta_{\beta}\int_Q\beta\overline{c}v_{\beta} \leq C\int_Q(\zeta_{\beta}-\zeta)^2 +C\beta^2\int_Q\zeta_{\beta}^2. \] All the terms of the form above $\int_Q(\zeta_{\beta}-\zeta)^2$ and $\int_Q(\chi_{\beta}-\chi)^2$ can be combined with the $c_1\int_Q(\zeta_{\beta}-\zeta)^2$ and $c_2\int_Q(\chi_{\beta}-\chi)^2$ in equation (\ref{sdeqn}). Terms above with $\zeta_{\beta}^2$, $\chi_{\beta}^2$ are estimated as follows: \begin{eqnarray*} \lefteqn{ \left(\int_Q\zeta_{\beta}^2\right)\left(\int_Q(v_{\beta}-v)^2\right) + \left(\int_Q\chi_{\beta}^2\right)\left(\int_Q(u_{\beta}-u)^2\right) }\\ &\leq &\left(\|u_{\beta}-u\|^2_{L^2(Q)}\| + \|v_{\beta}-v\|^2_{L^2(Q)}\|\right) \left(\|\zeta_{\beta}\|^2_{L^2(Q)} + \|\chi_{\beta}\|^2_{L^2(Q)}\right). \end{eqnarray*} Other terms include \[ C\beta^2\int_Q\zeta_{\beta}^2,\,\, C\beta^2\int_Q\chi_{\beta}^2 \] and \[ C\int_Q(v_{\beta}-v)^2,\, C\int_Q(u_{\beta}-u)^2. \] Now use the estimate in equation (\ref{fuest}) to get \[ \left(\|u_{\beta}-u\|^2_{L^2(Q)} + \|v_{\beta}-v\|^2_{L^2(Q)}\right) \leq C\beta^2\|\overline{c}\|^2_{L^2(Q\times Q)}. \] Using proposition (\ref{est}) with $\alpha = \bar{c}$ and $\beta = 0$ we derive \[ \left(\|\zeta_{\beta}\|^2_{L^2(Q)} + \|\chi_{\beta}\|^2_{L^2(Q)}\right) \leq C\|\overline{c}\|^2_{L^2(Q \times Q)}. \] The above estimates provide an {\it a priori} bound for the second derivative which proves (\ref{secdu}) and (\ref{secdv}). These weak convergences of the quotients justify that $\eta$ and $\tau$ satisfy the weak formulation of the system (\ref{secdeqn}). \hfill $\diamondsuit $ \\ {\bf Remark 1.} The estimates in the above Proposition also give us uniform $ L^2 $ bounds for the second derivatives of $\tau $ and $\eta $. Namely, \[ \|\tau\|_{L^2(Q)}+\|\eta\|_{L^2(Q)} \leq C\|\overline{c}\|^2_{L^2(Q\times Q)}.\] \\ {\bf Remark 2.} From the proof of previous proposition, \begin{equation} \begin{array}{lr} {\displaystyle \frac{u_{\beta} - u}{\beta}} \rightarrow \zeta & \mbox{strongly in } L^2(Q), \\[2ex] {\displaystyle \frac{v_{\beta} - v}{\beta}} \rightarrow \chi & \mbox{strongly in } L^2(Q), \\[2ex] {\displaystyle \frac{\zeta_{\beta} - \zeta}{\beta}} \rightarrow \tau & \mbox{strongly in } L^2(Q), \\[2ex] {\displaystyle \frac{\chi_{\beta} - \chi}{\beta}} \rightarrow \eta & \mbox{strongly in } L^2(Q). \end{array} \end{equation} We have a similar result for the second derivatives of $u$ and $v$ with respect to the control $d$. \begin{prop} The mapping \[ d \mapsto (u(c,d),v(c,d)) \in V^2 \] admits second derivatives with respect to $c$ in the sense \begin{equation} \label{secdu1} \begin{array}{lr} \displaystyle \frac{\zeta(c,d+\beta\overline{d},0;\overline{d})-\zeta(c,d;0,\overline{d})} {\beta} \rightarrow \kappa & \mbox{ weakly in } V, \end{array} \end{equation} \begin{equation} \label{secdv1} \begin{array}{lr} \displaystyle \frac{\chi(c,d + \beta\overline{d},0;\overline{d})-\chi(c,d;0,\overline{d})} {\beta} \rightarrow \delta & \mbox{ weakly in } V, \end{array} \end{equation} as $\beta \rightarrow 0$, for any $(c,d) \in [C_{\Gamma}]^2$ and $\overline{d} \in L^{\infty}(Q)$ such that $d+\beta\overline{d} \in C_{\Gamma}. $\\ Also $(\kappa,\delta) \stackrel{\rm def}{=}((\kappa(c,d;0,\overline{d}; 0,\overline{d}),\delta(c,d;0,\overline{d}; 0,\overline{d}))$ is the unique solution of \begin{equation} \label{secdeqn1} { {\cal L}} \left( \begin{array}{c} \kappa\\ \delta \end{array} \right) = -2\left( \begin{array}{c} \xi\int_Qd\sigma + v\int_Q\overline{d}\sigma + \xi\int_Q\overline{d}u \\ \sigma\int_Qc\xi \end{array} \right). \end{equation} with $\kappa = \delta = 0$ at $t=0$ and $\kappa = \delta = 0$ on $\Sigma$. \label{second1} \end{prop} \begin{prop}{For a fixed $d \in C_{\Gamma}, $ the mapping $c \in C_{\Gamma} \mapsto {\cal J}(c,d)$ is strictly convex.} \label{convex} \end{prop} {\bf Proof:} As mentioned earlier, it suffices to show that ${\mit J}''({\nu}) > 0$ for $\nu \in [0,1].$ \\[2ex] The justification for differentiating $\mit J$ is a consequence of the above established first and second derivatives of $u,v$ with respect to the control variable $c$ and the strong convergence noted in the previous proposition. %Also note that;\\[2ex] %$ \bullet \frac{u_{\beta}-u}{\beta} \rightarrow \tau $ strongly in $L^2(Q)$. %\\[2ex] %$ \bullet \frac{v_{\beta}-v}{\beta} \rightarrow \eta$ strongly in $L^2(Q)$. %\\[2ex] %This follows from an argument akin to the establishment of the solutions of %the state system.\\[2ex] Now, for $0 \leq \nu \leq 1$, \[ {\mit J}(\nu) = {\cal J}(\overline{c}+ \nu(c-\overline{c}), d).\] The directional derivative is now in the direction $c-\overline{c}$.\\ Denoting \[ u=u(\overline{c} + \nu(c-\overline{c}),d), \] \[ v=v(\overline{c} + \nu(c-\overline{c}),d), \] \[ \zeta =\zeta(\overline{c} + \nu(c-\overline{c}),d;c-\overline{c},0), \] \[ \chi =\chi(\overline{c} + \nu(c-\overline{c}),d;c-\overline{c},0), \] \[ \tau =\chi(\overline{c} + \nu(c-\overline{c}),d;c-\overline{c},0; c-\overline{c},0), \] \[ \eta =\eta(\overline{c} + \nu(c-\overline{c}),d;c-\overline{c},0; c-\overline{c},0), \] \[ {\mit J}(\nu) = \int_Q\left\{K[u-\tilde{u}]^2 - L[v-\tilde{v}]^2 \right\} + \int_Q\int_Q\left\{N(\overline{c}+\nu(c-\overline{c}))^2-Md^2\right\}. \] Differentiating twice with respect to $c$ \begin{eqnarray} {\mit J}''(\nu) & = & \int_Q (K\zeta^2 + K[u-\tilde{u}]\tau - L[v-\tilde{v}]\eta -L\chi^2) \nonumber\\ & & \mbox{}+ \int_Q \int_Q N(c-\bar{c})^2. \end{eqnarray} \begin{eqnarray*} {\mit J}''(\nu) & \geq & -K\|u\|_{L^2(Q)}\|\tau\|_{L^2(Q)} - K\|\tilde{u}\|_{L^2(Q)}\|\tau\|_{L^2(Q)} \\[2ex] & - & L\|\chi\|^2_{L^2(Q)} - L\|v\|_{L^2(Q)}\|\eta\|_{L^2(Q)}\\[2ex] & - & L\|\tilde{v}\|_{L^2(Q)}\|\eta\|_{L^2(Q)} + N\|c-\overline{c}\|^2_{L^2(Q)}. \end{eqnarray*} From proposition \ref{second}, for $\epsilon > 0$ there exists $c_0(\epsilon)$ such that for $c_{k} > c_0,\\ k = 1, 2,$ \[ \|\tau\|_{L^2(Q)} + \|\eta\|_{L^2(Q)} \leq \epsilon\|c-\overline{c}\|^2_{L^2(Q\times Q)}\] and \[ \|\chi\|^2_{L^2(Q)} \leq\epsilon\|c-\overline{c}\|^2_{L^2(Q\times Q)}.\] Combining these estimates with above we get \[ {\mit J}''(\nu) > (N-\tilde{\epsilon})\|c-\overline{c}\|^2_{L^2 (Q\times Q)}.\] \hfill $\diamondsuit $ \begin{prop}{ For a fixed $d \in C_{\Gamma}$ the mapping $c \in C_{\Gamma} \mapsto {\cal J}(c,d)$ is lower semicontinuous in the weak topology on $L^2(Q \times Q).$ } \label{lsc} \end{prop} {\bf Proof:} It is enough to show for every $\alpha \in$ {\bf R}, \[S(d,\alpha) = \{h | h \in C_{\Gamma}, {\cal J}(h,d) \leq \alpha \} \] is closed in the weak topology of $L^2(Q \times Q).$ Let $d \in C_{\Gamma}$ and $\alpha \in$ {\bf R} be fixed such that \[ {\cal J}(c_{n},d) \leq \alpha \] and \[ c_{n} \rightarrow \hat{c} \mbox{ weakly in } L^{\infty}(Q \times Q).\] Using the state systems, $u_{n}(c_{n},d),v_{n}(c_{n},d)$ satisfy \[ \|u_{n}\|_{V},\|v_{n}\|_{V} \leq C. \] Then, using compactness results \cite[Chapter 4, Prop. 4.2]{li2} , we can find subsequences such that \[ u_{n} \rightarrow u \hspace{10mm} \mbox{ weakly in } V, \mbox{ strongly in } L^2(Q) \] \[v_{n} \rightarrow v \hspace{10mm} \mbox{ weakly in } V, \mbox{ strongly in } L^2(Q). \] Standard continuity arguments show that \[ u=u(\hat{c},d) \hspace{10mm} v=v(\hat{c},d). \] Also using a generalization of Fatou's lemma, \[ \lim \inf {\cal J}(c_{n},d) \geq {\cal J}(\hat{c},d). \] \hfill $\diamondsuit $ % \begin{thm}{If $c_0$ is large enough, there exists a unique saddle point $(c^{\ast}, d^{\ast}).$} \end{thm} {\bf Proof: } Combining Propostion \ref{convex} and Proposition \ref{lsc} and the existence of saddle point result from Ekeland and Temam \cite[Chap.6, Propositions 1.5 and 2.1]{et}, we conclude that the cost functional admits a unique saddle point. \hfill $\diamondsuit $ \section{The Optimality System} The solution of the optimality system, consisting of the two state equations and two suitably chosen adjoint equations, will be used to characterize the saddle point of the game. \begin{thm}{If $(c,d) \in [C_{\Gamma}]^2$ is the saddle point and $ c_0 $ is sufficiently large, then there exists $(u,v,p,q) \in V^{4}$ satisfying: \begin{equation} \begin{array}{rlr} L_1u + u\int_Qcv & = f & \\[2ex] L_2v + v\int_Qdu & = g &\\[2ex] L_1^{\ast}p + p\int_Qcv - \int_Qd^Tvq & = K(u-\tilde{u}) & \\[2ex] L_2^{\ast}q + q\int_Qdu -\int_Qc^Tup & =-L(v-\tilde{u}) & \mbox{in }\, Q \end{array} \end{equation} \begin{equation} \begin{array}{rlr} u(x,t) = u^0, v(x,t) & = v^0 & \mbox{on }\, \Omega \times\{0\} \\[2ex] u(x,t) = v(x,t) & = 0 & \mbox{on }\, \Sigma \\[2ex] p(x,T) = q(x,T) &= 0 & \mbox{on }\, \Omega \times {T}\\[2ex] p(x,t) = q(x,t) &= 0 & \mbox{on}\, \Sigma, \end{array} \end{equation} where \[ L^{\ast}_{k}u = -u_t - (a^k_{ij}u_{x_j})_{x_i} - (b^k_iu)_{x_i} + c^ku,\; k =1, 2. \] Moreover on $ Q $, %$ c = {\displaystyle \min(\frac{p^{+}uv}{N},\Gamma)} \hspace{5mm} %d = {\displaystyle \min(\frac{q^{+}uv}{M},\Gamma)} \hspace{3mm} \mbox{on } Q.$ \[ c(x,y,t,\tau) = \min(\frac{p^{+}(x,t)u(x,t)v(y, \tau)}{N}, \Gamma),\] \[ d(x,y,t,\tau) = \min(\frac{q^{+}(x,t)u(y, \tau)v(x,t)}{M}, \Gamma), \] where $p^{+} = max(p,0)$.} \end{thm} {\bf Proof:} Let $(c,d)$ $\in$ $[C_{\Gamma}]^2$ be a saddle point. Choose $\overline{c}$ $\in$ $L^{\infty}(Q \times Q)$ in such a way that for $\beta > 0$ arbitrarily small, $c + \beta\overline{c}$ lies in the set $C_{\Gamma}$. Since $(c,d)$ is a saddle point, \begin{equation} \label{quotient} \lim_{\beta \to 0} \frac{{\cal J}(c + \beta\overline{c},d) - {\cal J}(c,d)} {\beta} \geq 0. \end{equation} % Substituting the explicit form of the cost functional in (\ref{quotient}), dividing by $\beta$, and noting $\frac{u_{\beta}-u}{\beta}\rightarrow \zeta$ strongly in $L^2(Q)$, and $u_{\beta}\rightarrow u$, we get \begin{equation} \label{quot} \int_Q (K(u-\tilde{u})\zeta - L(v-\tilde{v})\chi)\,dx\,dt + \int_Q \! \int_Q Nc\overline{c}\,dxdy\,dtd\tau\, \geq 0. \end{equation} We introduce a new notation: \[ d^T(x,y,t,\tau): = d(y,x,\tau,t),\, c^T(x,y,t,\tau) := c(y,x,\tau,t). \] Now we define an operator ${ {\cal L}}^{\ast}$ such that {\bf formally } \[ (p,q){{\cal L}}\left( \begin{array}{c} \zeta\\ \chi \end{array} \right ) = (\zeta, \chi){{\cal L}}^{\ast}\left(\begin{array}{c} p \\ q \end{array} \right). \] We define the adjoint functions $(p,q)$ as the solutions in $ V $ of \begin{equation} \label{adjoint} \displaystyle \begin{array}{rlr} L_1^{\ast}p + p\int_Q cv - \int_Q d^Tvq &= K(u-\tilde{u}),& \mbox{in }\, Q \\ L_2^{\ast}q + q\int_Q du - \int_Q c^Tup &= -L(v-\tilde{v}),& \\[2ex] p(x,T) = q(x,T) &= 0, & \mbox{on }\, \Omega \times {T}\\ p(x,t) = q(x,t) &= 0, & \mbox{on }\, \Sigma. \end{array} \end{equation} The solution of the above system, after a change of variable $\hat{p}(x,t)=p(x,T-t)$ and $\hat{q}(x,t)=q(x,T-t)$, is constructed in a manner similar to that of the solution of the original state system. % Substituting (\ref{adjoint}) in (\ref{quot}), we get \[ \int_Q (\zeta,\chi){{\cal L}}^{\ast} \left( \begin{array}{l} p \\ q \end{array} \right) +\int_Q \int_Q Nc\overline{c}\, \geq 0 \] % Now, from (\ref{firstsys1}) \[\int_Q(p,q){{\cal L}}\left(\begin{array}{c} \zeta \\ \chi \end{array} \right) + \int_Q\int_Q Nc\bar{c} = \int_Q (p,q) \left( \begin{array}{c} -u\int_Q \overline{c}v \\ 0 \end{array} \right) +\int_Q \int_Q Nc\overline{c}\, \geq 0 \] % This implies \begin{equation} \label{controlc} \int_Q \int_Q (Nc-puv)\overline{c} \geq 0. \end{equation} % Since we can choose $\bar{c}$ non-negative and arbitrary, this implies \[ Nc - puv \geq 0.\] On the set $\{(x,y,t,\tau) | c(x,y,t,\tau) = 0\}$ we get $puv \leq 0$ which gives \[ p^{+} = 0. \] On the set $\{(x,y,t,\tau) |0 < c(x,y,t,\tau) \Gamma \}, \bar{c}$ has arbitrary sign, which implies $p \geq 0$ and \[ c = \frac{p^{+}uv}{N}. \] On the set $\{(x,y,t,\tau) | c(x,y,t,\tau) = \Gamma \}, \bar{c}$ must be non-positive, which gives \[ Nc - puv \leq 0.\] Combining these results we get \[ c(x,y,t,\tau) = \min(\frac{p^{+}(x,t)u(x,t)v(y, \tau)}{N}, \Gamma)\]. \\ Similarly, \[ d(x,y,t,\tau) = \min(\frac{q^{+}(x,t)u(y, \tau)v(x,t)}{M}, \Gamma). \] \hfill $ \diamondsuit $ % \begin{thm}{ For $ c_0 $ sufficiently large, bounded solutions of the optimality system: \begin{equation} \begin{array}{rlr} L_1u + u\int_Q min(\frac{p^{+}uv}{N},\Gamma)v & = f & \\[2ex] L_2v + v\int_Q min(\frac{q^{+}uv}{M},\Gamma)u & = g &\\[2ex] L_1^{\ast}p + p\int_Q min(\frac{p^{+}uv}{N},\Gamma)v - \int_Q min(\frac{q^{+}uv}{M},\Gamma)^Tvq & = K(u-\tilde{u}) & \\[2ex] L_2^{\ast}q + q\int_Q min(\frac{q^{+}uv}{M},\Gamma)u -\int_Q min(\frac{p^{+}uv}{N},\Gamma)^Tup & =-L(v-\tilde{u}) & \mbox{in }\,Q \end{array} \end{equation} \begin{equation} \begin{array}{rlr} u(x,t) = u^0, v(x,t) & = v^0 & \mbox{on }\, \Omega \times{0} \\[2ex] u(x,t) = v(x,t) & = 0 & \mbox{on }\, \Sigma \\[2ex] p(x,T) = q(x,T) &= 0 & \mbox{on }\, \Omega \times {T}\\[2ex] p(x,t) = q(x,t) &= 0 & \mbox{on }\, \Sigma. \end{array} \end{equation} exist and are unique in the solution space $[V]^{4}$ .} \end{thm} {\bf Proof: } The existence of the saddle point implies the existence of $u, v$ and then the existence of $p, q$. The optimality system for a strictly convex-concave functional completely characterizes its saddle points \hfill $\diamondsuit $ \section{Summary} We have proved that a two person zero sum game described by a system of parabolic equations with competitive interactions can be controlled via the non-local kernels of the interacting terms, and the saddle point can be represented in terms of the optimality system. %\subsection*{Acknowledgement} %This paper is part of the author's paper under the direction %of Professor Suzanne Lenhart at the University of Tennessee. \begin{thebibliography}{99} \bibitem{et} Ekeland, I. and Temam, R. (1976) Convex Analysis and Variatioanal Problems. {\em North Holland, Amsterdam}. \bibitem{evans} Evans L.C. Partial Differential Equations. {\em Berkeley Mathematics Lecture Notes}, {\bf Vol. 3B}. \bibitem{f} Fife, P.C. (1979) Mathematical Aspects of Reaction-Diffusion Systems. {\em Lecture notes in Mathematics, vol 28.} Springer Verlag. \bibitem{l} Lancaster, F.W. (1914) Aircraft in warfare: The dawn of the fourth Arm--No.V : The principle of concentration. {\em Engineering} {\bf 98}, 422-423. \bibitem{lp} Lenhart, S. and Protopopescu, V.(1993) Optimal Control for Semilinear Nonlocal Competitive Systems. {\em Appl. Math. Optim.} {\bf 28}, 113-132. \bibitem{li1} Lions, J.L. (1961) Optimal control of systems governed by Partial differential equations. {\em Springer Verlag,} New York. \bibitem{li2} Lions, J.L. (1961) Equations Differentielles Operationelles et Problems aux limites. {\em Springer Verlag,}. \bibitem{lps} Lenhart, S., Protopopescu, V. and Stojanovic, S. (1993) A minmax problem for semilinear nonlocal competitive systems.{\em Appl. Math. Optim.} {\bf 28}, 113-132. \bibitem{ps} Protopopescu, V. and Santoro, R.T. (1989) Combat modeling with partial differential equations. {\em European Journal of Optim. Res.} {\bf 38} 178-183. \bibitem{sp} Szpiro, A.A. , Protopopescu, V. (1993) Parabolic systems with competitive interactions and control on intial conditons. {\em Applied Math. and Computation} {\bf 59} 215 - 245. \end{thebibliography} \medskip \noindent{\sc Sanjay Chawla}\\ Department of Computer Science \\ University of Minnesota \\ Minneapolis, MN 55455 USA\\ e-mail: chawla@cs.umn.edu \end{document}