A new proof to the necessity of a second moment stability condition of discrete-time Markov jump linear systems with real states (Q442998)

Summary: We study the second moment stability of a discrete-time jump linear system with real states and the system matrix switching in a Markovian fashion. A sufficient stability condition was proposed by \textit{Y. Fang}, and \textit{K. A. Loparo} [``Stochastic stability of jump linear systems'', IEEE Transactions on Automatic Control, vol. 47, no. 7, 1204--1208, 2002] which only needs to check the eigenvalues of a deterministic matrix and is much more computationally efficient than other equivalent conditions. The proof to the necessity of that condition, however, is a challenging problem. In the paper by \textit{M. D. Fragoso}, \textit{O. L.V. Costa} [''Mean square stabilizability of continuous-time linear systems with partial information on the Markovian jumping parameters'', Stochastic Anal. Appl. 22, No. 1, 99-111 (2004; Zbl 1125.93055)], a proof was given by extending the state domain to the complex space. This paper proposes an alternative necessity proof, which does not need to extend the state domain. The proof in this paper demonstrates well the essential properties of the Markov jump systems and achieves the desired result in the real state space.