Game Dynamic Problems for Systems with Fractional Derivatives

Abstract

A general approach to solving game approach problems for systems with Volterra evolution is outlined. It is based on the method of resolving functions [11] (the latter is also referred to as the method of Minkowski inverse functionals [12]) and employs the apparatus of the theory of set-valued mappings. Suggested scheme encompasses a wide range of functional-differential systems, in particular integral, integro-differential, and difference-differential systems of equations that specify dynamics of the conflict-controlled process.In this chapter, we examine in great detail the case when dynamics of the conflictcontrolled process is described by a system with fractional derivatives. Note that we deal with both the Riemann–Liouville and Dzhrbashyan–Nersesyan–Caputo fractional derivatives.Here solutions to such systems are given in the form of Cauchy formula analog. Sufficient conditions for terminating the game of approach in some guaranteed time are obtained. These conditions are based on the Pontryagin condition analog [48], expressed in terms of the Mittag–Leffler matrix functions [13,14]. Using the asymptotic expansions of these functions allows one to develop conditions for solvability of the game problems.