On Dynamical Optimization of Conflict Hereditary Systems

Abstract

For dynamical systems with aftereffect, the problem of control under disturbances is considered within the game-theoretic approach of N.N. Krasovskii and A.I. Subbotin. The problem is posed in the class of strategies with memory (functions of the motion history). The value of optimal guaranteed result (OGR) depends here on an initial history. An appropriate functional equation of the Hamilton[Jacobi]Bellman[Isaacs (HJBI) type with co-invariant (ci-) derivatives is presented. It is shown that if the functional of OGR is ci-smooth then it is the classical solution of this equation, and the optimal strategy can be constructed by aiming in the direction of its ci-gradient. In the nonsmooth case, a generalization of the presented HJBI equation is obtained by using an appropriate directional derivatives. Here, for constructing the optimal control strategy, the method of aiming in the direction of ci-gradients of auxiliary Lyapunov type functionals is elaborated.