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.

在标准线性二次 (LQ) 控制中，研究无限范围最优控制的第一步是用最优 LQ 控制器推导出稳定条件。本文着重于成本泛函中具有不确定控制和状态加权矩阵的 Itô 随机系统的稳定性。广义代数Riccati方程(GARE)是通过广义微分Riccati方程(GDRE)在有限视界情况下的收敛得到的。更重要的是，得到了不确定随机控制的充分必要稳定条件。关键技术之一是将 GARE 的解分解为满足奇异代数 Riccati 方程 (SARE) 的半正定矩阵和作为满足某些线性矩阵不等式条件的集合的元素的常数矩阵。使用 GARE 和 SARE 之间的等价性，我们将一般不定情况的稳定性降低到确定情况的稳定性，其中使用由受 SARE 约束的最优成本函数定义的 Lyapunov 函数来研究稳定性。