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In standard linear quadratic (LQ) control, the first step in investigatinginfinite-horizon optimal control is to derive the stabilization condition withthe optimal LQ controller. This paper focuses on the stabilization of an Itostochastic system with indefinite control and state weighting matrices in thecost functional. A generalized algebraic Riccati equation (GARE) is obtainedvia the convergence of the generalized differential Riccati equation (GDRE) inthe finite-horizon case. More importantly, the necessary and sufficientstabilization conditions for indefinite stochastic control are obtained. One ofthe key techniques is that the solution of the GARE is decomposed into apositive semi-definite matrix that satisfies the singular algebraic Riccatiequation (SARE) and a constant matrix that is an element of the set satisfyingcertain linear matrix inequality conditions. Using the equivalence between theGARE and SARE, we reduce the stabilization of the general indefinite case tothat of the definite case, in which the stabilization is studied using aLyapunov functional defined by the optimal cost functional subject to the SARE.

Stabilization Control for ItO Stochastic System with Indefinite State and Control Weight Costs