Optimal control for parabolic systems with competitive interactions

Abstract

Communicated by L. PayneWe consider a two-dimensional parabolic system with general competitive interactions as a two-player game with conflicting objectives and with controls on the inhomogeneous (source) terms. We show the existence of an optimal solution of the game as the saddle point of a suitable objective functional. IntroductionWe consider a system of two semilinear parabolic equations with general competitive interactions. These systems occur frequently in mathematical modelling as suitable representations of chemical, biological, ecological, military or economic situations. Specifically, we consider the competitive system as a two-player zero-sum game which means that one player's success leads necessarily to the other player's loss. Each player (competitor) has control over some of the game parameters and tries to use these parameters in order to modify the evolution of the system in a desired or prescribed direction and to achieve his own objectives. The explicitly conflicting objectives of the players, are usually modelled in terms of minimizing (respectively maximizing) a certain functional. This functional-that depends on the state of the system and on the controls-gives a quantitative measure of the players' reaching their objectives.Of course, a game has meaning only if one can prove that these objectives are reachable. The interest in application of control theory for second-order partial differential equations and/or systems, stirred by the seminal work of Lions The purpose of this paper is to show that a rather general class of two-players competitive game described by semilinear parabolic equations can be controlled through the inhomogeneous terms (external sources) to a unique saddle point. We show that the solution, which is the saddle point of the objective functional is represented by the solution of the Optimality System (0s). The optimality system