Minimax and viscosity solutions in optimization problems for hereditary systems

For a dynamical system with discrete and distributed time delays, a control problem under disturbance or counteraction is considered. The problem is formalized in the context of the game-theoretical approach in the class of control strategies with memory. The problem is associated with a functional Hamilton-Jacobi type equation with coinvariant derivatives. The minimax and viscosity approaches to a generalized solution of this equation are discussed. It is shown that, under the same condition at the right endpoint, the minimax and viscosity solutions coincide, thereby uniquely defining the functional of optimal guaranteed result in the control problem.