FUNCTIONAL HAMILTON-JACOBI TYPE EQUATIONS WITH CI-DERIVATIVES IN CONTROL PROBLEMS WITH HEREDITARY INFORMATION

Abstract

In this paper the problem of control of dynamic systems complicated by aftereffect is considered under conditions of uncertain but bounded disturbances. The problem is formalized as the differential game with hereditary information (in the class of pure strategies with memory, as functions of the history of motion). The so-called co-invariant (ci-)derivatives of the value functional of this game are considered and the functional Hamilton-Jacobi-Bellman-Isaacs type equation in terms of these derivatives is presented. It is shown that in the case when the value functional is ci-smooth it is the classical solution of this equation and the solving optimal strategies can be constructed by aiming in direction of its ci-gradient. In general (nonsmooth) case it is shown that the value functional is the generalized (minimax) solution of the given equation; and here, for construction of solving strategies, the method of aiming in direction of ci-gradients of some auxiliary ci-smooth functionals is developed.