Weak detectability and the LQ problem of discrete-time infinite Markov jump linear systems

The paper deals with a concept of weak detectability for discrete-time infinite Markov jump linear systems, which relates the stochastic convergence of the output with the stochastic convergence of the state and generalizes previous concepts. Certain invariant sets are introduced, which allow us to find a related system that is stochastically detectable if and only if the original system is weakly detectable. This provides the necessary tools to show, via an additional assumption, that the weak detectability concept is invariant with respect to linear state feedback control. As an immediate extension, the result provides that linear state feedback controls are stabilizing whenever the associated cost functional is bounded. In addition, it is shown that the detectability concept assures that the solution of the JLQ is stabilizing and the solution of the associated algebraic Riccati equation is unique and stabilizing, thus retrieving the usual role that detectability concepts play in finite dimensional MJLS and linear deterministic systems. Finally, regarding the assumption, the paper shows that: it is not related to the detectability concept, it always holds for finite dimensional Markov jump linear systems, and it holds under a condition of uniform observability on trajectories associated with non-convergent output.