Solvability of the Main Constrained Problem and Some Extensions

Abstract

This chapter is devoted to the study of the problems of the existence an optimal control strategy and of the form of that strategy. The solvability of the main convex programming problem (2.2) is proved for the semiconti-nuous models, and the different versions of Markov decision processes are explicitly investigated. Under some conditions the form of an optimal control strategy is established: that is, the mixture of (N + 1) selectors (Markov or stationary selectors in the corresponding models). If one investigates a (homogeneous) Markov model then it is sufficient to consider only Markov (stationary) randomized strategies. The set of Markov selectors is sufficient for the convex Markov models. The elementary example illustrates all the theoretical reasonings.