Differential games problems described by system of infinite differential equations in Hilbert space

This thesis deals with the solutions of differential game problems described by some infinite systems of ordinary differential equations in Hilbert space. The infinite system arises from the solution of some control and differential game of problems described by some partial differential equations. By using decomposition method, some of these problems can be reduced to the one described by some infinite system of ordinary differential equations. Therefore, this thesis focuses on different types of infinite systems using various approaches in Hilbert space. The first system is an infinite system of first order differential equations, and the second system is an infinite system of 2-systems of first order differential equations. For all the systems, we study the existence and uniqueness, and then we consider control and differential game problems with some forms of constraints on controls of the players. For the first system, we present solution of optimal pursuit problems with negative coefficients, where the controls of the players are subjected to integral constraints. Pursuer’s goal is to force the state of the system toward the origin and the evader tries to avoid this. Secondly, we extend the first system and introduce another state away from that of the initial state. In this game, pursuer attempts to bring the state of the system toward another the evader’s purpose is opposite where we study pursuit game problems with negative coefficients. Furthermore, the second game is improved with various constraints and the coefficients assumed to be any real numbers, the condition of completion of pursuit with geometric and integral constraints is proposed. For the second system, we solve pursuit differential game problem of 2-system of first-order that involves a generalization of all considered games with conjugate complex, the case of integral constraints. The main findings and contributions of this thesis is to study differential game described by infinite system of differential equations. For the first system, we propose an optimal pursuit time. For the second the third cases, we propose a new approach of completion of pursuit and for the second system, a guaranteed pursuit time is also proposed.

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