On the Theory of Optimal Processes in Discrete Systems

In this paper, by introducing the notion of $\gamma$-convex set, we isolate a wider class of discrete control systems in which the global maximum principle holds. A new type of variation of control for such classes of discrete control systems is proposed and stronger global maximum principle and second-order optimality condition expressed in terms of a singular control of new type are obtained. Generalizing the notion of the relative interior of sets, we obtain an optimality condition for discrete systems in the form of an equality, which we call Pontryagin's equation.