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

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.