Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations

Abstract

The paper deals with a zero-sum differential game in which the dynamical system is described by a fractional differential equation with the Caputo derivative of an order α ∈ (0, 1). The goal of the first (second) player is to minimize (maximize) the value of a given quality index. The main contribution of the paper is the proof of the fact that this differential game has the value, i.e., the lower and upper game values coincide. The proof is based on the appropriate approximation of the game by a zero-sum differential game in which the dynamical system is described by a first order functional differential equation of a retarded type. It is shown that the values of the approximating differential games have a limit, and this limit is the value of the original game. Moreover, the optimal players' feedback control procedures are proposed that use the optimally controlled approximating system as a guide.

Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

“…Optimal control problem: statement for an arbitrary position. Thus, in accordance with section 4 (see also [13,15]), by a position of system (3.1), we mean a pair (t, w(•)) consisting of a time t ∈ [0, T ] and a function w…”

“…Let us note that this approach goes back to the control theory for functionaldifferential systems. For more details on the relationship between fractional-order systems and functional-differential systems of a neutral type, the reader is referred to, e.g., [15,Sect. 4].…”

“…Thus, we conclude that, in the optimal control problem under consideration, the value (the optimal result) should be introduced as a functional ρ(t, w(•)), where the function w(τ ), τ ∈ [0, t], is treated as a history of a motion of the system on [0, t]. This circumstance significantly differs the approach developed in the paper from the existing works on the dynamic programming principle for fractional-order systems (see, e.g., [18,36,37]), in which the value is a function ρ(t, x), where x is treated as a value of the state vector at the time t. Let us note that this approach, involving the dependence of the value on a history of a motion, is generally accepted in the control theory for functional-differential systems, and, therefore, due to the relationship between fractional-order systems and functional-differential systems of a neutral type (see, e.g., [15,Sect. 4]), it seems natural to use this approach in the considered problem, too.…”

“…On the other hand, we prove that if the Cauchy problem for this equation and a natural right-end condition admits a ci-smooth of the order α solution, then it coincides with the value functional, and, moreover, we can construct an optimal feedback control strategy by using the extremal shift in the direction of the ci-gradients of the order α of this solution. Here, we use a quite general notion of feedback control strategies that goes back to the positional approach in differential games [28,26] (see also [33,29,30,31] and [13,15]). Finally, we illustrate the obtained results by an example.…”

Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

“…In particular, the authors of [15][16][17][18] consider the problems of pursuit of two persons with fractional derivatives. In [19], a proof is given of the existence of the prices of the game in a differential game described by an equation with fractional derivatives. The evader-pursuit problem with phase restrictions with fractional derivatives of the order α ∈ (0, 1) is addressed in [20,21].…”