Analysis of Noise under Regime Switching

ABSTRACT

In this paper we consider a stochastic nonlinear system under regime switching. Given a system*x(t)=f(x(t),r(t),t)* in which f satisfies so-called one-side polynomial growth condition. We introduce two Brownian noise feedbacks and stochastically perturb this system into *dx(t)=(x(t),r(t),t)dt+ σ (r(t))|x(t)|*^{β}x(t)dW_{1}(t)+q(r(t))x(t)dW_{2}(t) . It can be proved that appropriate noise intensity may suppress the potentially explode in a finite time and ensure that this system is almost surely exponentially stable although the corresponding system without Brownian noise perturbation may be unstable system.

In this paper we consider a stochastic nonlinear system under regime switching. Given a system

KEYWORDS

Hybrid System, One-Side Polynomial Growth Condition, Ito Formula, Stochastic Ultimate Boundedness

Hybrid System, One-Side Polynomial Growth Condition, Ito Formula, Stochastic Ultimate Boundedness

Cite this paper

nullL. Bai and X. Li, "Analysis of Noise under Regime Switching,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 836-842. doi: 10.4236/am.2011.27112.

nullL. Bai and X. Li, "Analysis of Noise under Regime Switching,"

References

[1] R. Z. Hasminskii, “Stochastic Stability of Differetial Equations,” Sijthoff and Noordhoof, Alphen, 1980.

[2] X. Mao, “Exponential Stability of Stochastic Differential Equations,” Dekker, New York, 1994.

[3] X. Mao, “Stochastic Differential Equations and Their Applications,” Horwood Publishing, Chichester, 1997.

[4] X. Mao, “Stability and Stabilizition of Stochastic Differetnial Delay Equations,” Proceeding of IET on Control and Theory and Application, November 2007, pp. 1551-1566. doi: 10.1049/iet-cta:20070006

[5] J. A. D. Appleby, X. R. Mao and A. Rodkina, “Stabilization and Destabilization of Nonlinear Differential Equations by Noise,” IEEE Transactions on Automatic Control, Vol. 53, No. 3, 2008, pp. 683-691. doi:10.1109/TAC.2008.919255

[6] X. Mao, G. Marion and E. Renshaw, “Environmental Brownian Noise Suppresses Explosions in Population Dynamics,” Stochastic Processes and Their Applications, Vol. 97, No. 1, 2002 pp. 95-110. doi:10.1016/S0304-4149(01)00126-0

[7] F, Q. Deng, Q. Luo; X. R. Mao and S. L. Pang, “Noise Suppresses or Expresses Exponential Growth,” Systems Control Letters, Vol. 57, No. 3, 2008. doi:10.1016/j.sysconle.2007.09.002

[8] X. Mao and C. Yuan, “Stochastic Differential Equations with Markovian Switching,” Imperial College Press, London, 2006.

[9] X. R. Mao; G. G. Yin and C. G. Yuan, “Stabilization and Destabilization of Hybrid Systems of Stochastic Differential Equations,” Vol. 43, No. 2, 2007, pp. 264-273. doi:10.1016/j.automatica.2006.09.006

[10] G. D. Hu, M. Z. Liu, X. R. Mao and M. H. Song, “Noise Expresses Exponential Growth under Regime Switching,” Systems Control Letters, Vol. 58, No. 9, 2009, pp. 691-699. doi:10.1016/j.sysconle.2009.06.006

[11] G. D. Hu, M. Z. Liu, X. R. Mao and M. H. Song, “Noise Suppresses Exponential Growth under Regime Switching,” Journal of Mathematical Analysis and Applications, Vol. 355, No. 2, 2009, pp. 783-795. doi:10.1016/j.jmaa.2009.02.009

[12] F. Wu and S. G. Hu, “Suppression and Stabilization of Noise,” International Journal of Control, Vol. 82, No. 11, 2009, pp. 2150-2157.

[13] C. Zhu and G. Yin, “Asymptotic Properties of Hybrid Diffusion Systems,” SIAM Journal on Control and Optimazation, Vol. 46, No. 4, 2007, pp. 1155-1179. doi:10.1137/060649343

[1] R. Z. Hasminskii, “Stochastic Stability of Differetial Equations,” Sijthoff and Noordhoof, Alphen, 1980.

[2] X. Mao, “Exponential Stability of Stochastic Differential Equations,” Dekker, New York, 1994.

[3] X. Mao, “Stochastic Differential Equations and Their Applications,” Horwood Publishing, Chichester, 1997.

[4] X. Mao, “Stability and Stabilizition of Stochastic Differetnial Delay Equations,” Proceeding of IET on Control and Theory and Application, November 2007, pp. 1551-1566. doi: 10.1049/iet-cta:20070006

[5] J. A. D. Appleby, X. R. Mao and A. Rodkina, “Stabilization and Destabilization of Nonlinear Differential Equations by Noise,” IEEE Transactions on Automatic Control, Vol. 53, No. 3, 2008, pp. 683-691. doi:10.1109/TAC.2008.919255

[6] X. Mao, G. Marion and E. Renshaw, “Environmental Brownian Noise Suppresses Explosions in Population Dynamics,” Stochastic Processes and Their Applications, Vol. 97, No. 1, 2002 pp. 95-110. doi:10.1016/S0304-4149(01)00126-0

[7] F, Q. Deng, Q. Luo; X. R. Mao and S. L. Pang, “Noise Suppresses or Expresses Exponential Growth,” Systems Control Letters, Vol. 57, No. 3, 2008. doi:10.1016/j.sysconle.2007.09.002

[8] X. Mao and C. Yuan, “Stochastic Differential Equations with Markovian Switching,” Imperial College Press, London, 2006.

[9] X. R. Mao; G. G. Yin and C. G. Yuan, “Stabilization and Destabilization of Hybrid Systems of Stochastic Differential Equations,” Vol. 43, No. 2, 2007, pp. 264-273. doi:10.1016/j.automatica.2006.09.006

[10] G. D. Hu, M. Z. Liu, X. R. Mao and M. H. Song, “Noise Expresses Exponential Growth under Regime Switching,” Systems Control Letters, Vol. 58, No. 9, 2009, pp. 691-699. doi:10.1016/j.sysconle.2009.06.006

[11] G. D. Hu, M. Z. Liu, X. R. Mao and M. H. Song, “Noise Suppresses Exponential Growth under Regime Switching,” Journal of Mathematical Analysis and Applications, Vol. 355, No. 2, 2009, pp. 783-795. doi:10.1016/j.jmaa.2009.02.009

[12] F. Wu and S. G. Hu, “Suppression and Stabilization of Noise,” International Journal of Control, Vol. 82, No. 11, 2009, pp. 2150-2157.

[13] C. Zhu and G. Yin, “Asymptotic Properties of Hybrid Diffusion Systems,” SIAM Journal on Control and Optimazation, Vol. 46, No. 4, 2007, pp. 1155-1179. doi:10.1137/060649343