On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems

Abstract

The paper is devoted to the development of the viscosity approach to the generalized solution of functional Hamilton-Jacobi type equations with coinvariant derivatives and a nonanticipatory Hamiltonian. These equations are naturally connected to problems of dynamical optimization of hereditary systems and, as compared with classical Hamilton-Jacobi equations, possess a number of additional peculiarities stipulated by the aftereffect. The definition of a viscosity solution that takes the above peculiarities into account is given. The consistency of this definition with the notion of a classical solution and with the minimax approach to the generalized solution is substantiated. The existence and uniqueness theorems are proved.