We study a deterministic linear-quadratic (LQ) control problem over an infinite horizon, without the restriction that the control cost matrix R or the state cost matrix Q be positive-definite. We develop a general approach to the problem based on semidefinite programming (SDP) and related duality analysis. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control under a certain stability condition (which is satisfied automatically when Q is positive-definite). When the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP.
Bibliographical noteFunding Information:
Manuscript received March 25, 1999; revised February 23, 2000 and October 24, 2000. Recommended by Associate Editor Y. Yamamoto. This work was supported in part by the National Science Foundation under Grant ECS-97-05392, in part by the Hong Kong RGC Earmarked Grants CUHK 4175/00E, CUHK 4181/00E, CUHK 4125/97E, and in part by two Direct Grants from the Chinese University of Hong Kong.
- Complementary duality
- Generalized Riccati equation
- LQ control
- Semidefinite programming