Minimax Solutions of Hamilton--Jacobi Equations with Fractional Coinvariant Derivatives

TL;DR

This paper develops a theory for Hamilton--Jacobi equations involving fractional coinvariant derivatives, proving existence, uniqueness, and consistency of minimax solutions in the context of fractional optimal control.

Contribution

It introduces a generalized minimax solution concept for fractional Hamilton--Jacobi equations and establishes key properties including a comparison principle.

Findings

Existence and uniqueness of minimax solutions are proven.

The minimax solution is shown to be consistent with classical solutions.

A comparison principle is established using a Lyapunov--Krasovskii functional.

Abstract

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by ordinary differential equations with the Caputo fractional derivatives of the order $\alpha$. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov--Krasovskii functional.