Linear Discrete Pursuit Game with Phase Constraints

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Position of the evader satisfies phase constraints: y ∈ G, where G is a subset of R n. We considered two cases: (1) controls of the players satisfy geometric constraints, and (2) controls of the players satisfy total constraints. Terminal set M is a subset of R n and it is assumed to have a nonempty interior. Game is said to be completed if y(k) − x(k) ∈ M at some step k; thus, the evader has not the right to leave set G. To construct the control of the pursuer, at each step i, we use the value of the control parameter of the evader at the step i. We obtain sufficient conditions of completion of pursuit from certain initial positions of the players in finite time interval and construct a control for the pursuer in explicit form.


Introduction
A number of works were devoted to investigate differential and discrete pursuit games with various constraints on controls (see, e.g., [1][2][3][4][5][6][7][8]). Differential games with phase constraints on a position of one or several players are studied in many works (see, e.g., [9][10][11][12][13][14][15][16][17][18]). Most of these works were devoted to studying simple motion pursuit-evasion game problem. For example, in [10], an evasion problem is solved, when the evader moves on a curve and the pursuer moves on the plane. In this work, it is assumed that the maximum speed of the pursuer is equal to 1, and the maximum speed of the evader is more than 1. In the work [11], a simple motion pursuit-evasion game problem of many pursuers and one evader is considered on a compact set. The main result of this paper is estimation for guaranteed pursuit time. The work [12] is devoted to studying a game problem on a convex closed set when controls of players satisfy total constraints. In [13], simple motion pursuit game problem is solved for all initial positions of space when the maximum speeds of players are equal and the evader moves on a convex bounded set with nonempty interior. In [15], sufficient conditions of completion of pursuit are obtained for a linear differential game when the evader moves in a bounded convex set.
In [16], the case, where the terminal and control sets are convex compact sets, is studied. In [17], necessary and sufficient conditions of solvability of the evasion problem in a convex polyhedron were received.
In the present paper, we consider a linear discrete pursuit game of one pursuer and one evader. We will study both total and geometric constraints on controls of players under assumption that the terminal set consists of an interior point in . The evader can move only in a given bounded convex set. Some sufficient conditions of completion of pursuit have been obtained.

Statement of the Problem
Consider a discrete game described by the equations where , , , V ∈ , ≥ 1, = 1, 2, . . . , is step number, , are constant square matrices of order , and and are control parameters of the pursuer and evader, respectively. Control parameters and are constructed in the form of or the total constraints The pursuer moves according to (1) with control parameter , and the evader moves according to (2) with control parameter . The purpose of the pursuer is inclusion realization: for some final step , and the evader's purpose is opposite. State of the evader is subjected to constraint ∈ , where is a bounded convex set in . It is assumed that the terminal set is a subset of and it has nonempty interior. The condition int ̸ = Ø implies that there are a number ℓ > 0 and vector ∈ to satisfy the inclusion ℓ ⊂ − + , where is the ball of radius 1 centered at the origin. Since is a bounded convex set, there exists a number > 0 such that ⊂ .
satisfy the inclusion (7) at some ≤ = ( 0 ), then we say that pursuit can be completed for steps.

The Case of Geometric Constraints
Assumption 6. Let the following conditions be satisfied: Theorem 7. Let Assumption 6 hold and, for the position 0 , at some step = ( 0 ), the inclusion holds true. Then pursuit can be completed in the games (1)-(4) from initial position 0 for ( 0 ) steps.

Theorem 9. Let Assumption 8 hold and for the position
holds true. Then pursuit can be completed in the games (1)-(4) from the initial position 0 for ( 0 ) steps.

The Case of Total Constraints on Controls.
Let us consider the games (1), (2) with total constraints (5)-(6) on controls.

4
The Scientific World Journal Assumption 10. Let the following conditions be satisfied: (1) = , Theorem 11. Let Assumption 10 hold, and for the position 0 let the inclusion be satisfied at some step = ( 0 ). Then pursuit can be completed in the games (1), (2), (5), and (6) from the initial position 0 for ( 0 ) steps.
Let us consider the games (29), (5), and (6). It is not difficult to verify that Theorems 11 and 12 give the following result.
As we can see from inequalities (30) and (31) that Theorems 7 and 9 (resp., Theorems 11 and 12 in case of total constraints on controls) are applicable to the game (29) for some values of and (resp., 1 and 1 ) that satisfy the inequality < (resp., 1 < 1 ).

Conclusion
We have obtained sufficient conditions of completion of pursuit in linear discrete games, when phase constraints are imposed on position of evader. We have applied the obtained results to concrete discrete pursuit game problems with phase constraint on a position of the evader. We have constructed an example, for which pursuit problem is solvable even if speed (resource, in case of total constraints) of the pursier is a little bit less than the speed (resource) of the evader.