A primal-dual semi-definite programming approach to linear quadratic control

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 semi-definite 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.