On the Solvability of the Pursuit Problem for Differential Games with Fractional Hilfer Derivatives

Abstract

In the early 1960s, seminal results in differential game theory, where the game is described by an ordinary differential equation in a finite-dimensional space, were obtained by academicians L.S. Pontryagin and N.N. Krasovsky. In the 21st century, differential games described by fractional differential equations have been actively studied. In recent years, fractional calculus has gained a strong position in the mathematical modeling of physical, economic, and applied problems. Numerous examples demonstrate the successful application of ordinary differential equations and partial differential equations with fractional derivatives. In particular, the works of A.A. Chikrii, M.Sh. Mamatov, and E.M. Mukhsinov consider the pursuit problem when the game is described by fractional differential equations. In this paper, we investigate the solvability of the pursuit problem for a differential game with Hilfer fractional derivatives in a Banach space of order ????, 0 < ???? < 1, and type ????, 0 6 ???? 6 1. Using Pontryagin’s first method and the strict separation theorem, we prove two theorems providing sufficient conditions for the solvability of the pursuit problem and for the optimality of the pursuit time. A game problem describing the relaxation process during glass formation in supercooled liquids is solved.