Pursuit differential games of infinite three-dimensional system of differential equations in Hilbert space - Madhavan, Diviekga Nair (2023)
Early research on game problems described by a system of partial differential equations is followed by a reduction to those described by an infinite system of ordinary differential equations using the method of decomposition. Every infinite n-system of ordinary differential equations, n ≥ 2 has a solution with a unique fundamental matrix which is then applied to study differential games in various perspectives. This thesis focuses in finding solutions to pursuit differential game problems of an infinite 3-system of first order ordinary differential equations in Hilbert space l2. The model of the game is first formulated and then rewritten in a matrix form. The homogeneous solution of the model is obtained where a fundamental matrix is identified. Some notable properties of the fundamental matrix are proved and applied to find the particular solution of the model and simplify the calculations in the study of the differential game. The existence and uniqueness of the general solution of the game model in l2 space are then proved. The study of pursuit game begins with the problem of one pursuer and one evader where the pursuer aims to bring the state of the system from an initial state to the origin. On the other hand, the evader tries to prevent this from occurring as it moves freely. The game is studied separately with two different types of constraints on players’ control functions, which are integral and geometric constraints. The control problem is studied where the control function is first constructed and then shown to be admissible. The control function is to transfer the state of the system into origin and to be applied in construction of admissible strategy for the pursuer. Sufficient conditions are obtained for pursuer to complete the pursuit in a finite time interval. This thesis also examines pursuit differential games of both integral and geometric constraints where the pursuer’s motive is to transfer an initial non zero state of the system into another non zero state. This investigation also requires the control problem to be solved so that it can be used to establish an admissible strategy for the pursuer to bring the system to another non zero state within a finite time interval. A more refined study is carried out to solve optimal pursuit problem of the game with integral constraints where the evader moves with its own strategy rather than moving freely. In this investigation, an optimal control function is constructed and proven to be admissible. It is then utilised in establishing optimal strategies for both pursuer and evader to achieve the optimal pursuit time of the game. The final part of this thesis is about a study of pursuit game that involve finitely many pursuers versus one evader with model of the game is similar to the model of the previous games. The control function of each player is subjected to integral constraint. It is assumed that the combined resources of all pursuers is greater than the resource of the evader. An admissible strategy for each pursuer is constructed where two cases are considered. The first case is to show that the game of pursuit can be terminated by one of the pursuers at some time in a finite time interval in which the evader moves freely. The second case is to find the optimal number of pursuers needed to terminate the game in which the evader moves with constructed admissible strategy.