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

We study a deterministic linear quadratic LQ control problem over an in nite horizon without the restriction that the control cost matrix R or the state cost matrix Q be positive de nite We develop a general approach to the problem based on semi de nite programming SDP and related duality analysis We show that the complementary duality condition of the SDP is necessary and su cient for the existence of an optimal LQ control under certain stability condition which is satis ed automatically when Q is positive de nite When the complementary duality does hold an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP