SCIENCE CHINA Information Sciences, Volume 61, Issue 7: 070215(2018) https://doi.org/10.1007/s11432-017-9340-4

Suppression of explosion by polynomial noise for nonlinear differential systems

More info
  • ReceivedSep 24, 2017
  • AcceptedJan 8, 2018
  • PublishedJun 4, 2018


In this paper, we study the problem of the suppression of explosion by noise fornonlinear non-autonomous differential systems.For a deterministic non-autonomous differential system${\rm~d}{\boldsymbol~x}(t)={\boldsymbol~f}({\boldsymbol~x}(t),t){\rm~d}t$, which can explode at a finite time,we introduce polynomial noise and study the perturbed system${\rm~d}{\boldsymbol~x}(t)={\boldsymbol~f}({\boldsymbol~x}(t),t){\rmd}t~+h(t)^{\frac{1}{2}}|{\boldsymbol~x}(t)|^{\beta}A{\boldsymbol~x}(t){\rm~d}B(t)$.We demonstrate that the polynomial noise can not only guarantee the existence and uniqueness of theglobal solution for the perturbed system, but can also make almost every path of the global solutiongrow at most with a certain general rate and even decay with a certain general rate(including super-exponential, exponential, and polynomial rates) under specific weak conditions.


[1] Khasminskii R Z. Stochastic Stability of Differential Equations. Netherlands: Sijthoff and Noordhoff, 1981. 253--263. Google Scholar

[2] Arnold L, Crauel H, Wihstutz V. Stabilization of linear systems by noise. SIAM J Control Opt, 1983, 21: 451-461 CrossRef Google Scholar

[3] Mao X R. Exponential Stability of Stochastic Differential Equations. New York: Marcel Dekker, 1994. 167--171. Google Scholar

[4] Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Chichester: Horwood Publishing, 2007. 135--141. Google Scholar

[5] Appleby J A D, Mao X R. Stochastic stabilisation of functional differential equations. Syst Control Lett, 2005, 54: 1069-1081 CrossRef Google Scholar

[6] Appleby J A D, Mao X R, Rodkina A. Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans Automat Contr, 2008, 53: 683-691 CrossRef Google Scholar

[7] Mao X R. Stability and stabilisation of stochastic differential delay equations. IET Control Theor Appl, 2007, 1: 1551-1566 CrossRef Google Scholar

[8] Mao X R, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes Appl, 2002, 97: 95-110 CrossRef Google Scholar

[9] Bahar A, Mao X R. Stochastic delay Lotka-Volterra model. J Math Anal Appl, 2004, 292: 364-380 CrossRef Google Scholar

[10] Mao X R, Yuan C G, Zou J Z. Stochastic differential delay equations of population dynamics. J Math Anal Appl, 2005, 304: 296-320 CrossRef ADS Google Scholar

[11] Wu F K, Hu S G. Suppression and stabilisation of noise. Int J Control, 2009, 82: 2150-2157 CrossRef Google Scholar

[12] Song Y F, Yin Q, Shen Y. Stochastic suppression and stabilization of nonlinear differential systems with general decay rate. J Franklin Institute, 2013, 350: 2084-2095 CrossRef Google Scholar

[13] Yang Q Q, Zhu S, Luo W W. Noise expresses exponential decay for globally exponentially stable nonlinear time delay systems. J Franklin Institute, 2016, 353: 2074-2086 CrossRef Google Scholar

[14] Mao X R. Almost sure exponential stabilization by discrete-time stochastic feedback control. IEEE Trans Automat Contr, 2016, 61: 1619-1624 CrossRef Google Scholar

[15] Deng F Q, Luo Q, Mao X R. Noise suppresses or expresses exponential growth. Syst Control Lett, 2008, 57: 262-270 CrossRef Google Scholar

[16] Liu L, Shen Y. Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition. Automatica, 2012, 48: 619-624 CrossRef Google Scholar

[17] Feng L C, Wu Z H, Zheng S Q. A note on explosion suppression for nonlinear delay differential systems by polynomial noise. Int J General Syst, 2018, 47: 137-154 CrossRef Google Scholar

[18] Pavlovi? G, Jankovi? S. Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay. J Comput Appl Math, 2012, 236: 1679-1690 CrossRef Google Scholar

[19] Wu F K, Hu S G. Stochastic suppression and stabilization of delay differential systems. Int J Robust Nonlinear Control, 2011, 21: 488-500 CrossRef Google Scholar

[20] Fang S Z, Zhang T S. A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab Theor Relat Fields, 2005, 132: 356-390 CrossRef Google Scholar

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1       京公网安备11010102003388号