Weak controllability and weak stabilizability concepts for linear systems with Markov jump parameters

The paper introduces weak controllability and weak stabilizability concepts for discrete-time Markov jump linear system with finite Markov space. We introduce a collection of matrices Copf that resembles controllability matrices of deterministic linear systems. The collection of matrices Copf allows us to define a weak controllability concept by requiring that the matrices are full rank, as well as to introduce a weak stabilizability concept that is a dual of the weak detectability concept found in the literature of Markov jump systems. An important feature of the introduced concept is that it generalizes the concept of mean square stabilizability. The role that this concept plays in the scenario of the filtering problem is investigated through case studies, which suggest that weak stabilizability together with mean square detectability ensure that the state estimator is mean square stable. Illustrative examples are included