Linear Quadratic Control Revisited: A View Through Semidefinite Programming

We present a unified approach to both deterministic and stochastic linear-quadratic (LQ) control via the duality theory of semi-definite programming (SDP). This new framework allows the control cost matrix to be singular or even indefinite (in the stochastic setting), a useful feature in applications such as the optimal portfolio selection of financial assets. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control under certain stability conditions. When the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP. Furthermore, if thestrictcomplementarity holds, then a new optimal feedback control, which is always stabilizing, is generated via the dual SDP. On the other hand, for cases where the complementary duality fails and the LQ problem has no attainable optimal solution, we develop an e-approximation scheme that achieves asymptotic optimality.