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Solution Of A Singular Zero-Sum Linear-Quadratic Differential Game By Regularization

Author

Listed:
  • 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)

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.

Suggested Citation

  • Josef Shinar & Valery Y. Glizer & Vladimir Turetsky, 2014. "Solution Of A Singular Zero-Sum Linear-Quadratic Differential Game By Regularization," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1-32.
    • Handle: RePEc:wsi:igtrxx:v:16:y:2014:i:02:n:s0219198914400076
    DOI: 10.1142/S0219198914400076
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    References listed on IDEAS

    as
    1. Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
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    More about this item

    Keywords

    Singular differential game; regularization; cheap control; minimizing control sequence; 49N70; 91A23; 93C70;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

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