We consider the linear quadratic Gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon. Based on recent results from the deterministic framework, we characterise the feasibility of this problem using a linear matrix inequality. In particular, conditions for existence and uniqueness of optimal controls are derived, which are weaker compared to the standard approaches in the literature. We further show that also for the stochastic problem, the optimal control is given in terms of the stabilising solution of the Lur'e equation, which generalises the algebraic Riccati equation. We conclude our paper by examining an application of our theory to fluid dynamics problems.

我们考虑无限时间范围内描述系统的具有折扣成本函数的线性二次高斯控制问题。基于确定性框架的最新结果，我们使用线性矩阵不等式来描述该问题的可行性。特别是，导出了最优控制的存在条件和唯一性，与文献中的标准方法相比，这些条件较弱。我们进一步表明，对于随机问题，最优控制是根据 Lur'e 方程的稳定解给出的，该方程推广了代数 Riccati 方程。我们通过检验我们的理论在流体动力学问题中的应用来结束我们的论文。