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.
Highlights
IntroductionThe paper follows the positional approach (see, e.g., [1,2,3]) and is concerned with the questions of how to characterize the value functional and construct optimal positional (feedback) strategies of the players in 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 given Bolza-type cost functional
The paper follows the positional approach and is concerned with the questions of how to characterize the value functional and construct optimal positional strategies of the players in 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 given Bolza-type cost functional
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)
Summary
The paper follows the positional approach (see, e.g., [1,2,3]) and is concerned with the questions of how to characterize the value functional and construct optimal positional (feedback) strategies of the players in 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 given Bolza-type cost functional. Using standard arguments (see, e.g., [3] (Section 11.5) and [17] for time-delay systems), it is proved that this solution coincides with the value functional of the original differential game, and, optimal positional strategies of the players can be obtained by applying the extremal aiming in the direction of the ci-gradient of the order α of this solution. In a general non-smooth case, following, e.g., [3], it is shown that the value functional coincides with a generalized minimax solution of the given Cauchy problem [18,19] The proof of this fact reduces to construction of optimal positional strategies of the players on the basis of this minimax solution.
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