Pursuit and evasion differential games described by infinite two-systems of first order differential equations

Differential games are a special kind of problems for dynamic systems particularly for moving objects. Many reseachers had drawn interests on control and differential game problems described by parabolic and hyperbolic partial differential equations which can be reduced to the ones described by infinite systems of ordinary differential equations by using decomposition method. The main purpose of this thesis is to study pursuit and evasion differential game problems described by first order infinite two-systems of differential equations in Hilbert space, Ɩ₂. The control functions of the players are subjected to the geometric constraints. Pursuit is considered completed, if the state of the system coincides with the origin. In the game, the pursuer’s goal is to complete the pursuit while oppositely, the evader tries to avoid this. First, we solve for first order non-homogenous system of differential equations to obtain the general solution zk(t), k = 1,2, .... Then, to validate the existence and uniqueness of the general solution, we first prove that the general solution exists in Hilbert space. Next, we prove that the general solution is continous on time interval [0;T]. Our main contribution is that we examine the game by solving an auxiliary control problem, validating a control function and find a time for which state of the system can be steered to the origin. Then, we solve pursuit problem by constructing pursuit strategy and obtain guaranteed pursuit time, θ₁, under the speed of pursuer Σk=1|uk(t,v(t))|² < p² for any v(.) Ɛ S(á). However, for evasion differential game problem, we prove that evasion is possible when the speed of evader, á, is greater or equal than that of pursuer, p, on the interval [0,T].

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