Detectability of Markov jump linear systems, the LQ problem and the coupled Riccati equations

In this paper, different detectability conditions that appear in the literature of Markov jump linear systems are collected and compared. The role that those conditions play in the associated jump linear quadratic problem is clarified, as they appear in this paper as necessary and sufficient conditions for stability and positivity of solutions. Along this line, we show that weak detectability is the weakest condition that relates the mean square stability of the state trajectory and the finiteness of the associated cost functional. This result, in addition to a characterization developed here for the kernel of the solution allows us to show that the solution stabilizes the system if and only if the system is weakly detectable. We also show that weak observability is not only a sufficient condition, but also a necessary one, for the solution to be positive definite. The results are extended to the associated coupled algebraic Riccati equation in a simple manner, as far as its minimal solution is identified with the solution of the linear quadratic problem. Illustrative numerical examples and comparisons are included.