Long-Time Behavior of Stochastic Linear-Quadratic Optimal Control Problems

The turnpike property refers to the phenomenon that in many optimal control problems over finite but long-time horizon, optimal trajectories approach to a steady state of the system and stay close to it for the major part of the time horizon. In the past several decades, the turnpike properties have attracted extensive attentions in control theory. Numerous results have been established for deterministic optimal control problems of both finite and infinite dimensions. However, the study of turnpike phenomena for stochastic optimal control is quite lacking in literature. This paper is concerned with the turnpike properties for stochastic linear-quadratic optimal control problems. Under suitable conditions, the strong exponential, the strong integral, and the mean-square turnpike properties are established. The crucial issues are to correctly formulate the corresponding static optimization problem and find the correction processes, which illustrate the essential differences between stochastic and deterministic cases.