Autonomous linear output of the Markov chain stabilization by square criterion on an infinite horizon

The solution of the linear output of the stochastic differential system optimal control problem on an infinite horizon is adapted for one particular case of indirect observation. The ergodic Markov chain plays the role of a dynamic system and the autonomous linear output formed by it provides indirect noisy observations on the state of the chain. The control purpose is formulated as output stabilization in positions determined by the chain and periodically changing with chain state changes. The solution, as in a similar problem with complete information, is obtained as the limit form of optimal control in the corresponding problem with a finite horizon. Sufficient conditions for the control existence turn out to be typical conditions for linear-quadratic problems of structure of the optimal Wonham filter, only linear components are present in the control and all nonlinearity is limited by the equation of the filtering estimate. Due to this, the existence conditions include only the requirements for the autonomous Riccati equation solution. A numerical experiment for the mechanical drive model used in previous studies is discussed. The purpose of the experiment is to show the difference in the use of optimal control and its autonomous version.