An Approach for Solving Discrete Game Problems with Total Constraints on Controls
We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal set is a subset of and it is assumed to have nonempty interior. Game is said to be completed if at some step . To construct the control of the pursuer, at each step , we use positions of the players from step 1 to step and the value of the control parameter of the evader at the step . We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed.
A large number of works are devoted to differential games where the position of the players changes continuously in time (see, e.g., [1–18]). Zero-sum differential games were first considered in the book of Isaacs  who derived the main equation of the theory of differential games. Though the Isaacs method is not complete (since the main equation may not have classical solutions or may have infinitely many generalized solutions ) he effectively applied it to solve many interesting game problems.
Krasovskiĭ and Subbotin  and Pontryagin  proposed two different formalizations for differential games. According to Pontryagin we have to identify ourselves with the pursuer in pursuit games and with the evader in evasion games. Many approaches have been proposed in literature to solve differential game problems under the integral constraints (see, e.g., [1–4, 6, 8, 9, 11–15, 17]).
In the present paper, we study a linear pursuit game with total constraints on controls. Such constraints are discrete analogues of integral constraints for differential games. There are a few papers that study discrete games under total constraints (see, e.g., [14–16, 19–22]).
In the linear discrete game studied by Satimov et al.  , the eigenvalues of the main matrix are assumed to be real. Some sufficient conditions of completion of pursuit were obtained in this paper. In the paper of Satimov et al. , motions of players are simple. It was a starting point for multiperson differential games with integral constraints. Some sufficient conditions were obtained for the game to be terminated.
In the paper of Ibragimov , a discrete game described by the equation is studied. Control of the evader satisfies total constraint. There are two game problems that are considered. In the first game, the control of the pursuer satisfies geometric constraint, and in the second it satisfies total constraint. Sufficient conditions of completion of game from any position were obtained.
Azamov and Kuchkarov  studied relationship between 0-controllability of linear discrete systems and completion of linear pursuit discrete game described by the equation The control of the pursuer is subjected to geometric constraint and that of the evader is subjected to total constraint. Necessary and sufficient conditions were obtained under which solvability of 0-controllability is equivalent to completion of pursuit.
Kuchkarov et al.  studied a discrete game whose position is described by the above equation. Different from the above game, both controls of the players are subjected to total constraints. They proved that if eigenvalues of the matrix in absolute value are less than 1 and , where is unit ball in centered at the origin, then pursuit can be completed from any initial position. This result much more improves the result of the work by Ibragimov and Kuchkarov .
In the present paper, we study a linear pursuit discrete game of one pursuer and one evader. It is assumed that total constraints are imposed on controls of players. The main point in the pursuit method is that, in one step, the pursuer forces the evader to expend a certain amount of his resources to prevent the game from being completed. Therefore, after finite times of such steps the resources of the evader will be exhausted, and then pursuit will be completed. We obtain sufficient conditions of completion of pursuit.
2. Statement of the Problem
In the Euclidian space , we consider a discrete game described by the following equations: where and are constant matrices, and (resp., ) is control parameter of the pursuer (evader). The parameters and are chosen in the form of sequences as follows: and subjected to the following constraints: where and are positive numbers. The pursuer and evader move according to (4) and (5), respectively. In the space , a terminal set , whose interior is not empty, is defined. The condition implies that there are a number and vector to satisfy the inclusion . The purpose of the pursuer is to realize the inclusion at some finite step , and that of the evader is opposite.
Definition 2. If for any admissible control of the evader one can construct an admissible control of the pursuer such that for the solutions
of (4) and (5) with the initial positions , where , , , and controls the inclusion (9) is satisfied at some , then we say that pursuit starting from the position can be completed in the game (4), (5) for steps. Here, we assume that the pursuer uses to construct , .
Problem. Find sufficient conditions, under which pursuit can be completed from any initial position , .
3. Main Result
In this section, we formulate and prove the main result of the paper. First of all, we state our basic assumption. Let the norm of the matrix be defined by .
Assumption 3. There exist numbers , , , , , a positive integer , and vector such that and(1) for any , , there exists a step such that (2)there exist linear operators , , such that for any the following inequalities hold true:(a), (b), ;(3) .Under this assumption we prove the following statement.
Proof. Let . Then it follows from condition (1) of Assumption 3 that there exists such that
Consider the equation
with respect to the unknown vector . According to (12), (13) has a solution. Let be the lexicographically minimum solution of (13).
We now construct a control for the pursuer. Let the evader use an arbitrary admissible control ; that is, behavior of the evader is any.
By Definition 2, the pursuer may use to construct on each step . Set while where is an integer. Then it is natural to consider the following three cases.
Case 1. The inequality (15) holds for all .
Case 2. At some , , the inequality sign in (15) turns to equality.
Case 3. At some , , the inequality (15) holds, but at it fails to hold and opposite inequality holds: It follows from (4) and (5) that for the solutions and corresponding to the initial position and control (14) we obtain In Case 1, since is a solution of (13), it follows from (17) at that Next, if pursuit is not completed in the game (4), (5) at the step , then in view of (18) we obtain Then using the Cauchy-Schwartz inequality and condition (5) (a) of Assumption 3 yields Hence, Clearly, if inequality (19) is not satisfied, then pursuit starting from the initial position , , is completed on the step in the game (4), (5). Therefore, assuming that pursuit is not completed up to the step in the game (4), (5), which implies inequalities (19) and (21), we conclude that the evader must expend the resources greater than for the steps .
In Case 2, the evader expends resources equal to for the steps .
We turn to Case 3. Since inequality (15) is satisfied at and (16) is satisfied at , then at some . Define the value of the pursuer’s control as follows: Admissibility of the pursuer’s control will be shown later. Thus, in Case 3, to guarantee that pursuit is not completed up to the step , the evader must expend resources greater than .
The process of the game up to the step in Case 1 and up to the step in Cases 2 and 3 will be referred to as the process of first approach of the pursuer to evader.
We now calculate the amount of resources expended by the pursuer on the process of the first approach.
In Case 1, using the Minkowski inequality, we obtain from (14) that In Case 2, in view of (14) we have In Case 3, we use (14) and (23) to estimate the resources of the pursuer. Since , using the Minkowski inequality, we see that Similarly, according to condition (5) (b) of Assumption 3 and the fact that , , the right-hand sides of inequalities (24)–(26) can be estimated from the above by . For example, the right-hand side of (24) is estimated as follows: Therefore,
We conclude from these estimations that if pursuit is not completed in the game (4), (5) during the process of first approach, the amount of resources expended by the evader is greater than or equal to , and that expended by the pursuer does not exceed .
Further, we continue as follows. In Case 1, the points will be taken as the initial position at , and in Cases 2 and 3, the points will be taken as the initial position at . We now consider (4), (5) with the initial position for the steps .
Given the initial position , , we find the least integer such that at some . Existence of such is guaranteed by condition (1) of Assumption 3.
Let , , be an arbitrary control of the evader. For the steps , we define the control of the pursuer by (14), with replaced by while the inequality (15) holds.
We apply the above argument again. If pursuit is not completed in the game (4), (5) during the process of second approach, then the amount of resources expended by the evader is not less than , and that expended by the pursuer is less than .
Now, -time repeated application of this reasoning enables us to conclude that at most at the th process of approach pursuit is completed in the game (4), (5).
Indeed, assuming the contrary we have which contradicts condition (3) of Assumption 3.
However for we have . The proof is complete.
Remark 5. Let condition of Assumption 3 hold for and . Then conditions and of Assumption 3 can be weakened. More precisely, we require the inclusion and drop condition . Then Theorem 4 guarantees that pursuit can be completed only from the initial points .
Example 6. Consider a discrete game described by equations where , . Let , , and
We have obtained sufficient conditions of completion of pursuit for a linear pursuit discrete game with total constraints where motions of the players are described by different-type linear equations. The control parameter of the pursuer is constructed based on .
It should be noted that conclusion of Theorem 4 is still true with appropriate changes if controls and are subjected to constraints and , . Further studies can be done to weaken conditions of Assumption 3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The present research was partially supported by the National Fundamental Research Grant Scheme (FRGS) of Malaysia, 01-01-13-1228FR.
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