Indefinite stochastic linear quadratic controls in infinite time horizon

This paper is concerned with an optimal stochastic linear-quadratic (LQ) control problem in an infinite time horizon, where the diffusion term in dynamics depends on both the state and control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem which may still be well-posed due to the deep nature of uncertainty involved. The problem gives rise to a generalized algebraic Riccati equation (GARE), which is however fundamentally different from the classical algebraic Riccati equation as a result of the indefiniteness of the LQ problem. To analyze the GARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the GARE. Moreover, we develop a computational approach to the GARE via a semidefinite programming associated with the LMIs.