Linear output control of Markov chain by square criterion. Complete information case

The problem of optimal control of the linear output of a stochastic differential system, formed by an additive jumping input, was solved. The goal of optimization is set by a quadratic criterion of a special type which allows one to formalize the tasks of tracking an abruptly changing target and stabilizing the system near the directions determined by the input. The problem is solved under the assumption that there is complete information, i. e., the known state of the input Markov chain. This statement complements the previously obtained solution of the problem with incomplete information, in which the control and estimation problems are separated, provided by the optimal in this case Wonham filter. The result obtained in the article, in addition to its independent significance, also provides a reference solution for analyzing the quality of control under conditions of indirect observations. The solution of the problem under consideration, as in the statement with incomplete information, is provided by the direct application of the dynamic programming method. The Bellman's equation is refined for a given input model — a martingale representation of the chain and a range of values limited by unit coordinate vectors are used. A numerical experiment was carried out, the results of which illustrate the efficiency of the obtained control algorithms in both settings, with complete information and indirect observations.