ANZIAM J. 49 (2007), no. 2, pp. 231–241.

Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching

Zhenting Hou Hailing Dong† Peng Shi
School of Mathematics
Central South University
Changsha
410075 Hunan
China		 School of Mathematics
Central South University
Changsha
410075 Hunan
China
hailing_fly@mail.csu.edu.cn		 Faculty of Advanced Technology
University of Glamorgan
Pontypridd CF37 1DL
UK
Received March 17, 2007; revised August 6, 2007

Abstract

In this paper, finite phase semi-Markov processes are introduced. By introducing variables and a simple transformation, every finite phase semi-Markov process can be transformed to a finite Markov chain which is called its associated Markov chain. A consequence of this is that every phase semi-Markovian switching system may be equivalently expressed as its associated Markovian switching system. Existing results for Markovian switching systems may then be applied to analyze phase semi-Markovian switching systems. In the following, we obtain asymptotic stability for the distribution of nonlinear stochastic systems with semi-Markovian switching. The results can also be extended to general semi-Markovian switching systems. Finally, an example is given to illustrate the feasibility and effectiveness of the theoretical results obtained.

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2000 Mathematics Subject Classification: primary 34A34; secondary 60K15
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2376???
†indicates author for correspondence

References

  1. O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters”, J. Math. Anal. Appl. 179 (1993) 154–178. MR1244955
  2. M. Davis, Markov models and optimization (Chapman and Hall, London, 1992). MR1283589
  3. R. J. Elliott and D. D. Sworder, “Control of a hybrid conditionally linear Gaussian processes”, J. Optim. Theory Appl. 74 (1992) 75–85. MR1168997
  4. X. Feng, K. A. Loparo, Y. Ji and H. J. Chizeck, “Stochastic stability properties of jump linear systems”, IEEE Trans. Automat. Control 37 (1992) 38–53. MR1139614
  5. W. Fleming, S. Sethi and M. Soner, “An optimal stochastic production planning problem with randomly fluctuating demand”, SIAM J. Control Optim. 25 (1987) 1494–1502. MR912452
  6. A. Jensen, A distribution model applicable to economics (Munksgaard, Copenhagen, 1954). MR63637
  7. Y. Ji and H. J. Chizeck, “Controllability, stabilizability and continuous-time Markovian jump linear-quadratic control”, IEEE Trans. Automat. Control 35 (1990) 777–788. MR1058362
  8. Y. Ji and H. J. Chizeck, “Jump linear quadratic Gaussian control: steady-state solution and testable conditions”, Control Theory Adv. Tech. 6 (1990) 289–319. MR1080011
  9. Y. Ji, H. J. Chizeck, X. Feng and K. A. Loparo, “Stability and control of discrete time jump linear systems”, Control Theory and Advanced Technology 7 (1991) 247–270. MR1117531
  10. H. Kushner, Stochastic stability and control (Academic Press, New York-London, 1967). MR216894
  11. M. F. Neuts, “Probability distributions of phase type”, in Liber Amicorum Prof. Belgium Univ. of Louvain, (1975) 173–206.
  12. M. F. Neuts, Matrix-geometric solutions in stochastic models (John Hopkins Univ. Press, Baltimore, Md., 1981). MR618123
  13. P. Shi and E. K. Boukas, “H_{∞} control for Markovian jumping linear systems with parametric uncertainty”, J. Optim. Theory Appl. 95 (1997) 75–99. MR1477351
  14. P. Shi, E. K. Boukas and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delays”, IEEE Trans. Automat. Control 44 (1999) 2139–2144. MR1735731
  15. P. Shi, E. K. Boukas and R. K. Agarwal, “Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters”, IEEE Trans. Automat. Control 44 (1999) 1592–1597. MR1707063
  16. P. Shi, E. K. Boukas, S. K. Nguang and X. Guo, “Robust disturbance attenuation for discrete-time active fault tolerant control systems with uncertainties”, Optimal Control Appl. Methods 24 (2003) 85–101. MR1971675
  17. A. V. Skorohod, Asymptotic methods in the theory of stochastic differential equations (American Mathematical Society, Providence, RI, 1989). MR1020057
  18. C. Yuan and X. Mao, “Asymptotic stability in distribution of stochastic differential equations with Markovian switching”, Stochastic Process. Appl. 103 (2003) 277–291. MR1950767
  19. C. Yuan, J. Zhou and X. Mao, “Stability in distribution of stochastic differential delay equations with Markovian switching”, System Control Lett. 50 (2003) 277–291. MR1950767
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