Optimization of Discrete and Differential Inclusions of Goursat-darboux Type Withstate Constraints

In the last decade, discrete and continuous time processes with lumped and distributed parameters found wide application in the field of mathematical economics and in problems of control dynamic system optimization and differential games [1–19]. The present article is devoted to an investigation of problems of this kind with distributed parameters, where the treatment is in finite-dimensional Euclidean spaces. It can be divided conditionally into four parts. In the first part (Section 2), a certain extremal problem is formulated for discrete inclusions of Goursat-Darboux type. For such problems we use constructions of convex and nonsmooth analysis in terms of convex upper approximations, local tents, and locally conjugate mappings for both convex and for nonconvex problems to get necessary and sufficient conditions for optimality. In the third part (Section 4), we use difference approximations of derivatives and grid functions on a uniform grid to approximate the problem with differential inclusions of Goursat-Darboux type and to formulate a necessary and sufficient condition for optimality for the discrete approximation problem. It is obvious that such difference problems can play an important role also in computational procedures. In the fourth part (Section 5), we are able to use results in Section 4 to get sufficient conditions for optimality for differential inclusions of Goursat-Darboux type. The derivation of this condition is implemented by passing to the formal limit as the discrete