On viscosity solutions of path-dependent Hamilton--Jacobi--Bellman--Isaacs equations for fractional-order systems

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for path-dependent Hamilton--Jacobi--Bellman--Isaacs equations related to fractional-order systems, specifically for a two-player zero-sum differential game with Caputo derivatives.

Contribution

It introduces a novel notion of viscosity solutions for fractional path-dependent PDEs and characterizes the value functional of the differential game as the unique solution.

Findings

Proves existence and uniqueness of viscosity solutions for fractional path-dependent HJB--Isaacs equations.

Characterizes the value functional as the unique viscosity solution.

Extends classical PDE methods to fractional and path-dependent settings.

Abstract

This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order $\alpha \in (0, 1)$ and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton--Jacobi--Bellman--Isaacs equation with so-called fractional coinvariant derivatives of the order $\alpha$ and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem.