In this chapter we define and characterize some fundamental control theory concepts, namely the stabilizability and the detectability notions, for a general class of systems described by impulsive stochastic linear differential equations. To the best of the authors knowledge, it seems that such structural properties for impulsive stochastic systems have not been considered yet in the literature. In Sect. 4.2, we define the concepts of stabilizability in mean square by linear state feedback as well as the concept of stabilizability in mean square by impulses in the state feedback form of the considered class of jump linear stochastic systems. Then, based on Lyapunov criteria for exponential stability in mean square provided in Sect. 3.4 we derive several necessary and sufficient conditions which will allow us to test the property of mean-square stabilizability by state feedback and mean-square stabilizability by impulses. These results are then specialized in Sect. 4.3 to the case of stochastic sampled-data controlled systems by introducing the concept of mean-square stabilizability by sample states measurements (see Definition 4.5). In Sect. 4.4, we define the concept of mean-square detectability of a jump linear stochastic system and we provide methods which allow us to decide if a given jump linear stochastic system is detectable or not. More precisely, we shall show how we may use the results from Sect. 3.4 in order to obtain necessary and sufficient conditions for mean-square detectability of the stochastic systems under consideration. The detectability result obtained in Sect. 4.4 will then be applied in Sect. 4.5 to the mean-square stability problem. The objective here is to propose a less conservative result than the one obtained in Theorem 3.2 . We show that under the detectability assumption a kind of mean-square stability of a jump stochastic linear differential equation of type ( 3.1 ) is achieved even in the case when the non-homogeneous backward GLJLDE ( 3.32 ) has the forcing term positive semidefinite instead of uniform positive definite and has a positive semidefinite global solution instead of a uniform positive one. Finally, Sect. 4.6 addresses the Duality between the concepts of mean-square stabilizability and mean-square detectability defined in Sects. 4.2 and 4.4, respectively.

Structural Properties of Jump Linear Stochastic Systems

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发表日期：

2025年

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Vasile Drăgan,
Samir Aberkane,
Ioan Lucian Popa

摘要

In this chapter we define and characterize some fundamental control theory concepts, namely the stabilizability and the detectability notions, for a general class of systems described by impulsive stochastic linear differential equations. To the best of the authors knowledge, it seems that such structural properties for impulsive stochastic systems have not been considered yet in the literature. In Sect. 4.2, we define the concepts of stabilizability in mean square by linear state feedback as well as the concept of stabilizability in mean square by impulses in the state feedback form of the considered class of jump linear stochastic systems. Then, based on Lyapunov criteria for exponential stability in mean square provided in Sect. 3.4 we derive several necessary and sufficient conditions which will allow us to test the property of mean-square stabilizability by state feedback and mean-square stabilizability by impulses. These results are then specialized in Sect. 4.3 to the case of stochastic sampled-data controlled systems by introducing the concept of mean-square stabilizability by sample states measurements (see Definition 4.5). In Sect. 4.4, we define the concept of mean-square detectability of a jump linear stochastic system and we provide methods which allow us to decide if a given jump linear stochastic system is detectable or not. More precisely, we shall show how we may use the results from Sect. 3.4 in order to obtain necessary and sufficient conditions for mean-square detectability of the stochastic systems under consideration. The detectability result obtained in Sect. 4.4 will then be applied in Sect. 4.5 to the mean-square stability problem. The objective here is to propose a less conservative result than the one obtained in Theorem 3.2 . We show that under the detectability assumption a kind of mean-square stability of a jump stochastic linear differential equation of type ( 3.1 ) is achieved even in the case when the non-homogeneous backward GLJLDE ( 3.32 ) has the forcing term positive semidefinite instead of uniform positive definite and has a positive semidefinite global solution instead of a uniform positive one. Finally, Sect. 4.6 addresses the Duality between the concepts of mean-square stabilizability and mean-square detectability defined in Sects. 4.2 and 4.4, respectively.