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

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.