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Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies

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  • Mikhail I. Gomoyunov

    (Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya Str., 16, 620108 Ekaterinburg, Russia
    Institute of Natural Sciences and Mathematics, Ural Federal University, Mira Str., 19, 620002 Ekaterinburg, Russia)

Abstract

The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α ∈ ( 0 , 1 ) and a Bolza-type cost functional. A relationship between the differential game and the Cauchy problem for the corresponding Hamilton–Jacobi–Bellman–Isaacs equation with fractional coinvariant derivatives of the order α and the natural boundary condition is established. An emphasis is given to construction of optimal positional (feedback) strategies of the players. First, a smooth case is studied when the considered Cauchy problem is assumed to have a sufficiently smooth solution. After that, to cope with a general non-smooth case, a generalized minimax solution of this problem is involved.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1667-:d:594992
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    References listed on IDEAS

    as
    1. Arkadii A. Chikrii, 2008. "Game Dynamic Problems for Systems with Fractional Derivatives," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Athanasios Migdalas & Leonidas Pitsoulis (ed.), Pareto Optimality, Game Theory And Equilibria, pages 349-386, Springer.
    2. 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.
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