  In standard linear quadratic (LQ) control, the first step in investigating
infinite-horizon optimal control is to derive the stabilization condition with
the optimal LQ controller. This paper focuses on the stabilization of an Ito
stochastic system with indefinite control and state weighting matrices in the
cost functional. A generalized algebraic Riccati equation (GARE) is obtained
via the convergence of the generalized differential Riccati equation (GDRE) in
the finite-horizon case. More importantly, the necessary and sufficient
stabilization 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 stabilization of the general indefinite case to
that of the definite case, in which the stabilization is studied using a
Lyapunov functional defined by the optimal cost functional subject to the SARE.
