Viscosity Solutions to Hereditary Dynamical Optimization*

Control under disturbances or counteraction is considered for dynamical systems with both discrete and distributed delays. The problem is posed within the game-theoretic approach of N.N. Krasovskii and A.I. Subbotin in the class of strategies with memory. An appropriate functional Hamilton–Jacobi–Bellman–Isaacs equation with co-invariant derivatives is associated with the control problem. It is shown that the functional of optimal guaranteed result (the value functional) is a viscosity-type solution of this equation. The uniqueness of such a viscosity solution is proved.