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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>is a subset of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>and it is assumed to have nonempty interior. Game is said to be completed if<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>y</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>x</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:mfenced><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:math>at some step<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>. To construct the control of the pursuer, at each step<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math>, we use positions of the players from step 1 to step<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math>and the value of the control parameter of the evader at the step<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math>. 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.

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