Autonomous differential system linear output control by square criterion on an infinite horizon

The problem of optimal control of the stochastic differential system linear output on an infinite horizon is solved. The solution is considered as the limit form of optimal control in the corresponding problem with a finite horizon. Sufficient conditions for the existence of control are given. They consist of the requirements of the stationarity of nonlinear dynamics, the finiteness of the quadratic target functional, the stabilizability of the linear output, and the existence of a limit in the Feynman–Katz formula describing the nonlinear part of control. The conditions for the linear part of the control are related to the classical results of the existence of a solution to the autonomous Riccati equation. The existence of a limit in the Feynman–Katz formula is associated with the solution of a parabolic equation that sets the coefficients for the nonlinear part of the control. A special case of linear drift is considered in which the nonlinear nature of the problem is preserved but optimal control turns out to be linear both in output and in the state variable. The results of a numerical experiment are presented which makes it possible to analyze the transient process in a problem with a finite horizon and an ergodic process in dynamics. For the control coefficients, the limiting transition to the optimal values of the corresponding optimal autonomous control is illustrated.