Stability, stabilizability and detectability for Markov jump discrete-time linear systems with multiplicative noise in Hilbert spaces

Abstract

In this article we discuss stability, stabilizability and detectability problems for Markov-jump discrete-time linear systems (MJDLSs) with multiplicative noise (MN) and countably infinite state space of the Markov chain. On the basis of a new solution representation formula, we give new deterministic characterizations of the stability and the detectability properties of MJDLSs with MN. These results are obtained using an operatorial approach and the properties of certain positive evolution operators defined on ordered Banach spaces of sequences of nuclear operators. Assuming detectability conditions and avoiding stochastic proofs, we prove that any global, nonnegative and bounded solution of the Riccati equation of control is stabilizing for the MJDLSs with MN and control. Finally, we apply our results to solve a linear quadratic optimal control problem. The theory is illustrated by an example.

Keywords:

• discrete-time linear systems
• Markov chains
• detectability
• stability
• optimal control

AMS Subject Classifications::

• 93Cxx
• 60J10
• 34Dxx
• 49N05