Selected pursuit and evasion differential game problems in Hilbert space

This thesis deals with the solution of some pursuit and evasion differential game problems described by some models in Hilbert space. The models arise from the solution of pursuit and evasion game problems described by some partial differential equations. Three different type of models are considered, where for each model, we solve pursuit and evasion problem with some forms of constraints on controls of the players. The first model is the infinite system of first order differential equations z˙k(t) + kz(t) = −uk(t) + vk(t), zk(0) = zk0, k = 1, 2, . . . , where zk, uk, vk, zk0 2 R1, z0 = (z10, z20, . . . ) 2 l2 r+1, u = (u1, u2, . . . ) is the control parameter of the pursuer, v = (v1, v2, . . . ) is that of the evader and 1, 2, . . . is a bounded sequence of negative numbers. For this model, we present solution of optimal pursuit problem, where the controls of the players are subjected to integral constraints. Secondly, we consider z˙k(t) + k(t)zk(t) = −uk(t) + vk(t), zk(0) = zk0, k = 1, 2, . . . , where z0 = (z10, z20, . . . ) 2 l2, k(t), k = 1, 2, . . . , are bounded, non-negative continuous functions such that k(0) = 0, k = 1, 2, . . . , on the interval [0, T] and all other parameters are defined as in the first model. In this case, we solve pursuit and evasion problems with integral, geometric, and mix constraints on control functions of the players. The third model is given by z¨k(t) + kz(t) = −uk(t) + vk(t), zk(0) = zk0, z˙k(0) = zk1, k = 1, 2, . . . , where zk, uk, vk 2 R1, k = 1, 2, . . . , z0 = (z10, z20, . . . ) 2 l2 r+1, z1 = (z11, z21, . . . ) 2 l2 r , u = (u1, u2, . . . ) is the control parameter of the pursuer and v = (v1, v2, . . . ) is the control parameter of the evader. Conditions for the solvability of pursuit and evasion problems described by this model are obtained. Furthermore, we also study control problems related to each of the three models. In the case of first and third models, necessary and sufficient conditions for which the state of the systems can be transfered to the origin are presented. Sufficient conditions are given for the control problem described by the second model for the cases of geometric and integral constraints on the control functions.

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