Stabilization control for Itô stochastic system with indefinite state and control weight costs

In standard linear-quadratic (LQ) control, the first step in investigating infinite-horizon optimal control is to derive the stabilisation condition with the optimal LQ controller. This paper focuses on the stabilisation of an Itô stochastic system with indefinite control and state-weighting matrices in the cost functional. A generalised algebraic Riccati equation (GARE) is obtained via the convergence of the generalised differential Riccati equation (GDRE) in the finite-horizon case. More importantly, the necessary and sufficient stabilisation conditions for indefinite stochastic control are obtained. One of the key techniques is that the solution of the GARE is decomposed into a positive semi-definite matrix that satisfies the singular algebraic Riccati equation (SARE) and a constant matrix that is an element of the set satisfying certain linear matrix inequality conditions. Using the equivalence between the GARE and SARE, we reduce the stabilisation of the general indefinite case to that of the definite case, in which the stabilisation is studied using a Lyapunov functional defined by the optimal cost functional subject to the SARE.