Suppression of explosion by polynomial noise for nonlinear differential systems

In this paper, we study the problem of the suppression of explosion by noise for nonlinear non-autonomous differential systems. For a deterministic non-autonomous differential system dx(t) = f(x(t), t)dt, which can explode at a finite time, we introduce polynomial noise and study the perturbed system dx(t) = f(x(t), t)dt + h(t) 12 |x(t)|ßAx(t)dB(t). We demonstrate that the polynomial noise can not only guarantee the existence and uniqueness of the global solution for the perturbed system, but can also make almost every path of the global solution grow 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]  Fuke Wu,et al.  Suppression and stabilisation of noise , 2009, Int. J. Control.

[2]  Hong-ke Wang,et al.  On the Exponential Stability of Stochastic Differential Equations , 2009, ICFIE.

[3]  Qi Luo,et al.  Noise suppresses or expresses exponential growth , 2008, Syst. Control. Lett..

[4]  Xuerong Mao,et al.  Stochastic differential delay equations of population dynamics , 2005 .

[5]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[6]  Xuerong Mao,et al.  Almost Sure Exponential Stabilization by Discrete-Time Stochastic Feedback Control , 2016, IEEE Transactions on Automatic Control.

[7]  Quan Yin,et al.  Stochastic suppression and stabilization of nonlinear differential systems with general decay rate , 2013, J. Frankl. Inst..

[8]  Tusheng Zhang,et al.  A study of a class of stochastic differential equations with non-Lipschitzian coefficients , 2005 .

[9]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[10]  Xuerong Mao,et al.  Stochastic stabilisation of functional differential equations , 2005, Syst. Control. Lett..

[11]  Fuke Wu,et al.  Stochastic suppression and stabilization of delay differential systems , 2011 .

[12]  Lichao Feng,et al.  A note on explosion suppression for nonlinear delay differential systems by polynomial noise , 2018, Int. J. Gen. Syst..

[13]  Hans Crauel,et al.  Stabilization of Linear Systems by Noise , 1983 .

[14]  Song Zhu,et al.  Noise expresses exponential decay for globally exponentially stable nonlinear time delay systems , 2016, J. Frankl. Inst..

[15]  Xuerong Mao,et al.  Stabilization and Destabilization of Nonlinear Differential Equations by Noise , 2008, IEEE Transactions on Automatic Control.

[16]  Svetlana Jankovic,et al.  Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay , 2012, J. Comput. Appl. Math..

[17]  X. Mao,et al.  Stability and stabilisation of stochastic differential delay equations , 2007 .

[18]  X. Mao,et al.  Exponential Stability of Stochastic Di erential Equations , 1994 .

[19]  Lei Liu,et al.  Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition , 2012, Autom..

[20]  Fen Yang,et al.  Stochastic Delay Lotka-Volterra Model , 2011 .