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Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations

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  • Mikhail Gomoyunov

    (Ural Branch of the Russian Academy of Sciences
    Ural Federal University)

Abstract

The paper deals with a zero-sum differential game in which the dynamical system is described by a fractional differential equation with the Caputo derivative of an order $$\alpha \in (0, 1).$$α∈(0,1). The goal of the first (second) player is to minimize (maximize) a given quality index. The main contribution of the paper is the proof of the fact that this differential game has the value, i.e., the lower and upper game values coincide. The proof is based on the appropriate approximation of the game by a zero-sum differential game in which the dynamical system is described by a first-order functional differential equation of a retarded type. It is shown that the values of the approximating differential games have a limit, and this limit is the value of the original game. Moreover, the optimal players’ feedback control procedures are proposed that use the optimally controlled approximating system as a guide. An example is considered, and the results of computer simulations are presented.

Suggested Citation

  • Mikhail Gomoyunov, 2020. "Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations," Dynamic Games and Applications, Springer, vol. 10(2), pages 417-443, June.
    • Handle: RePEc:spr:dyngam:v:10:y:2020:i:2:d:10.1007_s13235-019-00320-4
    DOI: 10.1007/s13235-019-00320-4
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/5561 is not listed on IDEAS
    2. Hajipour, Ahamad & Hajipour, Mojtaba & Baleanu, Dumitru, 2018. "On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 139-153.
    3. Jun Shen & James Lam, 2014. "State feedback control of commensurate fractional-order systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 45(3), pages 363-372.
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    Cited by:

    1. Mikhail I. Gomoyunov, 2021. "Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies," Mathematics, MDPI, vol. 9(14), pages 1-16, July.

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