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Zbl 1055.93077
Yao, David D.; Zhang, Shuzhong; Zhou, Xun Yu
Stochastic linear-quadratic control via semidefinite programming. (English)
[J] SIAM J. Control Optimization 40, No. 3, 801-823 (2001). ISSN 0363-0129; ISSN 1095-7138/e

Summary: We study stochastic linear-quadratic (LQ) optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). A central issue is the stability of the feedback control; and we show this can be effectively examined through the complementary duality of the SDP. Furthermore, we establish several implication relations among the SDP complementary duality, the (generalized) Riccati equation, and the optimality of the LQ control problem. Based on these relations, we propose a numerical procedure that provides a thorough treatment of the LQ control problem via primal-dual SDP: it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution. For the latter case, we develop an $\epsilon$-approximation scheme that is asymptotically optimal.

MSC 2000:: *93E20 Optimal stochastic control (systems)
90C25 Convex programming
93D15 Stabilization of systems by feedback

Keywords: stochastic LQ control; semidefinite programming; complementary duality; mean-square stability; generalized Riccati equation

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