Optimal pursuit time for a differential game in the Hilbert space l2.

e consider a two-person zero-sum pursuit-evasion differential game in the Hilbert space l2. The control functions of the players are subject to integral constraints. It is assumed that the control resource of the pursuer is greater than that of the evader. The pursuer tries to force the state of the system towards the origin of the space l2, and the evader tries to avoid this. We give a solution to the optimal pursuit problem for the differential game. More precisely, we obtain an equation for the optimal pursuit time and construct optimal strategies for the players in an explicit form. To prove the main result we solve a time-optimal control problem

The optimal pursuit problem reduced to an infinite system of differential equations

The optimal game problem reduced to an infinite system of differential equations with integral constraints on the players’ controls is considered. The goal of the pursuer is to bring the system into the zeroth state, while the evader strives to prevent this. It is shown that Krasovskii's alternative is realized: the space of states is divided into two parts so that if the initial state lies in one part, completion of the pursuit is possible, and if it lies in the other part, evasion is possible. Constructive schemes for devising the optimal strategies of the players are proposed, and an explicit formula for the optimal pursuit time is derived

On Optimality of Pursuit Time

We study a differential game described by an infinite system of differential equations with integral constraints on controls of players. This system is obtained by a parabolic equation by using decomposition method. We obtained an equation to find the optimal pursuit time and examined the series representing the left hand side of the equation. Moreover necessary and sufficient condition for convergence of the series is obtained

Sufficient conditions for evasion in a linear differential game.

We study a linear evasion differential game in R2. Control sets of players, the pursuer and the evader, are compact subsets of R2. The terminal set of the game is the origin. The game is considered to be completed if the state of the system, z(t), reaches the origin. If z(t) never reaches the origin, then we say that evasion is possible in the game. We obtained weaker conditions for evasion than conditions obtained by other researches. We give some illustrative examples which show the advantage of our conditions

Evasion from one pursuer in a Hilbert space

We study a differential game of one pursuer and one evader described by infinite systems of second order ordinary differential equations. Controls of players are subjected to geometric constraints. Differential game is considered in Hilbert spaces. We proved one theorem on evasion. Moreover, we constructed explicitly a control of the evader

On control problem for infinite system of differential equations of second order

We study a control problem described by infinite system of differential equations of second order in the space 2 r 1l+. Control parameter is subjected to integral constraint. Our goal is to transfer the state of the system from a given initial position to the origin for finite time. We obtained necessary and sufficient conditions on the initial positions for which the problem can be solved

Pursuit-evasion differential game with many inertial players

We consider pursuit-evasion differential game of countable number inertial players in Hilbert space with integral constraints on the control functions of players. Duration of the game is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the functional, and the evader tries to maximize it. In this paper, we find the value of the game and construct optimal strategies of the players

On a class of simultaneous pursuit games.

Let A and B be given convex closed bounded nonempty subsets in a Hilbert space H; let the first player choose points in the set A and let the second one do those in the set B. We understand the payoff function as the mean value of the distance between these points. The goal of the first player is to minimize the mean value, while that of the second player is to maximize it. We study the structure of optimal mixed strategies and calculate the game value

An evasion differential game described by an infinite system of 2-systems of second order.

We study a differential game of many pursuers described by infinite systems of second order ordinary differential equations. Controls of players are subjected to geometric constraints. Differential game is considered in Hilbert spaces. We say that evasion is possible if ||zi(t)||r+1 + ||z˙i(t)||r 6= 0 for all i = 1, ...,m, and t > 0; m is the number of pursuers. We proved one theorem on evasion. Moreover, we constructed explicitly a control of the evader