Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives^*

¹ N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia.
² Ural Federal University, Ekaterinburg, Russia.

^** Corresponding author: m.i.gomoyunov@gmail.com

Received: 2 December 2020
Accepted: 23 February 2022

Abstract

We consider a Cauchy problem for a Hamilton–Jacobi equation with coinvariant derivatives of an order α ∈ (0, 1). Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order α. 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.