SOLUTION OF A SINGULAR ZERO-SUM LINEAR-QUADRATIC DIFFERENTIAL GAME BY REGULARIZATION

Josef Shinar (), Valery Y. Glizer () and Vladimir Turetsky ()
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Josef Shinar: Technion — Israel Institute of Technology, Haifa 32000, Israel
Valery Y. Glizer: Department of Applied Mathematics, Ort Braude College of Engineering, P.O. Box 78, Karmiel 21982, Israel
Vladimir Turetsky: Department of Applied Mathematics, Ort Braude College of Engineering, P.O. Box 78, Karmiel 21982, Israel

International Game Theory Review (IGTR), 2014, vol. 16, issue 02, 1-32

Abstract: A linear-quadratic zero-sum singular differential game, where the cost functional does not contain the minimizer's control cost, is considered. Due to the singularity, the game cannot be solved either by applying the MinMax principle of Isaacs, or by using the Bellman–Isaacs equation method. In this paper, the solution of the singular game is obtained by using an auxiliary differential game with the same equation of dynamics and with a similar cost functional augmented by an integral of the square of the minimizer's control multiplied by a small positive weighting coefficient. This auxiliary game is a regular cheap control zero-sum differential game. For the analysis of such a cheap control differential game, in the present paper a singular perturbation technique is applied. Based on this analysis, the minimizing control sequence and the maximizer's optimal strategy in the original (singular) game are derived. Moreover, the existence of the value of the original game is established and its expression is derived. The solution is illustrated by an interception example.

Keywords: Singular differential game; regularization; cheap control; minimizing control sequence; 49N70; 91A23; 93C70 (search for similar items in EconPapers)
JEL-codes: B4 C0 C6 C7 D5 D7 M2 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1142/S0219198914400076

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