Minimax solutions of Hamilton-Jacobi equations with fractional coinvariant derivatives*

We consider a Cauchy problem for a Hamilton-Jacobi equation with coinvariant derivatives of an order alpha is an element of (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 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.