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In this paper we study the optimization of the discrete-time stochastic linear-quadratic (LQ) control problem with conic control constraints on an infinite horizon, considering multiplicative noises. Stochastic control systems can be formulated as Markov Decision Problems (MDPs) with continuous state spaces and therefore we can apply the direct-comparison based optimization approach to solve the problem. We first derive the performance difference formula for the LQ problem by utilizing the state separation property of the system structure. Based on this, we successfully derive the optimality conditions and the stationary optimal feedback control. By introducing the optimization, we establish a general framework for infinite horizon stochastic control problems. The direct-comparison based approach is applicable to both linear and nonlinear systems. Our work provides a new perspective in LQ control problems; based on this approach, learning based algorithms can be developed without identifying all of the system parameters.

Optimization of Constrained Stochastic Linear-Quadratic Control on an Infinite Horizon: A Direct-Comparison Based Approach