Journal of Applied Mathematics
Volume 2012 (2012), Article ID 642480, 10 pages
Research Article

A New Proof to the Necessity of a Second Moment Stability Condition of Discrete-Time Markov Jump Linear Systems with Real States

1Department of Automation, University of Science and Technology of China, Anhui, Hefei 230027, China
2National Network New Media Engineering Research Center, Institute of Acoustics, Chinese Academy of Science, Beijing 100190, China

Received 12 February 2012; Accepted 28 March 2012

Academic Editor: Baocang Ding

Copyright © 2012 Qiang Ling and Haojiang Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies 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 Fang and Loparo (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 Costa and Fragoso (2004), 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.