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

## TL;DR

This paper proves that a zero-sum differential game with fractional dynamics has a well-defined value by approximating it with classical differential games and demonstrates the convergence of their values.

## Contribution

It introduces a method to establish the value of a fractional differential game through approximation by retarded-type differential games and constructs optimal feedback controls.

## Key findings

- The game has a well-defined value.
- Approximate game values converge to the original game value.
- Optimal feedback controls are constructed based on the 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 $\alpha \in (0, 1).$ 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.