This paper is devoted to the application of the recently developed results about exact detectability of a large class of time-varying hybrid stochastic systems (Drăgan et al., 2020). More specifically, we will show from one side the connection between this new detectability notion and the derivation of a criterion for exponential stability in mean square for this class of systems. In the second part, we will show the central role played by exact detectability in Riccati equations arising in linear quadratic (LQ) regulator problems for such stochastic systems: it is a necessary and sufficient condition for the Riccati equation to have a unique positive semi-definite and stabilizing solution.

本文致力于应用最近开发的关于一大类时变混合随机系统的精确可检测性的结果（Drăgan 等，2020）。更具体地说，我们将从一个方面展示这种新的可检测性概念与此类系统的均方指数稳定性标准的推导之间的联系。在第二部分中，我们将展示在此类随机系统的线性二次 (LQ) 调节器问题中出现的 Riccati 方程中精确可检测性所起的核心作用：Riccati 方程具有唯一的正半数是必要和充分条件。确定和稳定的解决方案。