Linear output control of Markov chains by the quadratic criterion

The problem of optimal output control of a stochastic observation system, in which the state determines an unobservable Markov jump process and linear observations are given by a system of Ito differential equations with a Wiener process, is solved. Observations additively include control vector, so that a controlled output of the system is formed. The optimization goal is set by a general quadratic criterion. To solve the control problem, a separation theorem is formulated that uses the solution to the optimal filtering problem provided by the Wonham filter. As a result of the separation, an equivalent problem of output control of a diffusion process of a particular type, namely, with linear drift and nonlinear diffusion, is formed. The solution of this problem is provided by direct application of the dynamic programming method.