Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Stability of random-switching systems of differential equations


Authors: C. Zhu, G. Yin and Q. S. Song
Journal: Quart. Appl. Math. 67 (2009), 201-220
MSC (2000): Primary 60J27, 60J75, 93D05, 93D20, 93E15
DOI: https://doi.org/10.1090/S0033-569X-09-01092-8
Published electronically: February 18, 2009
MathSciNet review: 2514632
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.


References [Enhancements On Off] (What's this?)

  • 1. G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, Journal of Finance, 42 (1987), 301-320.
  • 2. G.K. Basak, A. Bisi, and M.K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622. MR 1406250 (97g:60091)
  • 3. P. Billingsley, Convergence of Probability Measures, J. Wiley & Sons, Inc., New York, 1968. MR 0233396 (38:1718)
  • 4. M.H.A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46(3) (1984), 353-388. MR 790622 (87g:60062)
  • 5. J. Hale, Ordinary Differential Equations, 2nd Ed., R.E. Krieger Pub. Co., Malabar, FL, 1980. MR 587488 (82e:34001)
  • 6. I.Ia. Kac and N.N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mech., 24 (1960), 1225-1246.
  • 7. Y. Ji and H.J. Chizeck, Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automatic Control, 35 (1990), 777-788. MR 1058362 (91h:49037)
  • 8. R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980. MR 600653 (82b:60064)
  • 9. R.Z. Khasminskii, C. Zhu, and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 177-194.
  • 10. H.J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd Ed., Springer-Verlag, New York, NY, 2003. MR 1993642 (2004e:62005)
  • 11. X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67. MR 1666831 (2000e:60095)
  • 12. M. MARITON, Jump Linear Systems in Automatic Control, Marcel Dekker, Inc., New York, 1990.
  • 13. L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd Ed., New York, 2001. MR 1801796 (2001k:34001)
  • 14. A.V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989. MR 1020057 (90i:60038)
  • 15. G. Yin, and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit, IEEE Trans. Automat. Control, 50 (2005), 577-593. MR 2141561 (2006b:93245)
  • 16. G. Yin, V. Krishnamurthy, and C. Ion, Regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization, SIAM J. Optim., 14 (2004), 1187-1215. MR 2112970 (2005m:62148)
  • 17. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer-Verlag, New York, 1998. MR 1488963 (2000a:60142)
  • 18. G. Yin and C. Zhu, On the notion of weak stability and related issues of hybrid diffusion systems, Nonlinear Anal.: Hybrid Sys., 1 (2007), 173-187.
  • 19. X.Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. MR 2044805 (2004m:91130)
  • 20. C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process Appl., 103 (2003), 277-291. MR 1950767 (2003k:60142)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 60J27, 60J75, 93D05, 93D20, 93E15

Retrieve articles in all journals with MSC (2000): 60J27, 60J75, 93D05, 93D20, 93E15


Additional Information

C. Zhu
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: zhu@uwm.edu

G. Yin
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: gyin@math.wayne.edu

Q. S. Song
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: qingshus@usc.edu

DOI: https://doi.org/10.1090/S0033-569X-09-01092-8
Keywords: Random switching, hybrid system, stability, necessary condition, sufficient condition
Received by editor(s): September 6, 2007
Published electronically: February 18, 2009
Additional Notes: Research of the first author was supported in part by the National Science Foundation under DMS-0304928
Research of the second author was supported in part by the National Science Foundation under DMS-0603287, and in part by the National Security Agency under MSPF-068-029
Research of the third author was supported in part by the U.S. Army Research Office MURI grant W911NF-06-1-0094 at the University of Southern California
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society