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Suppression of explosion by polynomial noise for nonlinear differential systems

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Abstract

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.

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References

  1. 1

    Khasminskii R Z. Stochastic Stability of Differential Equations. Netherlands: Sijthoff and Noordhoff, 1981. 253–263

  2. 2

    Arnold L, Crauel H, Wihstutz V. Stabilization of linear systems by noise. SIAM J Control Optim, 1983, 21: 451–461

  3. 3

    Mao X R. Exponential Stability of Stochastic Differential Equations. New York: Marcel Dekker, 1994. 167–171

  4. 4

    Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Chichester: Horwood Publishing, 2007. 135–141

  5. 5

    Appleby J A D, Mao X R. Stochastic stabilisation of functional differential equations. Syst Control Lett, 2005, 54: 1069–1081

  6. 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

  7. 7

    Mao X R. Stability and stabilisation of stochastic differential delay equations. IET Control Theor Appl, 2007, 1: 1551–1566

  8. 8

    Mao X R, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes Appl, 2002, 97: 95–110

  9. 9

    Bahar A, Mao X R. Stochastic delay Lotka-Volterra model. J Math Anal Appl, 2004, 292: 364–380

  10. 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

  11. 11

    Wu F K, Hu S G. Suppression and stabilisation of noise. Int J Control, 2009, 82: 2150–2157

  12. 12

    Song Y F, Yin Q, Shen Y, et al. Stochastic suppression and stabilization of nonlinear differential systems with general decay rate. J Franklin Institute, 2013, 350: 2084–2095

  13. 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

  14. 14

    Mao X R. Almost sure exponential stabilization by discrete-time stochastic feedback control. IEEE Trans Automat Contr, 2016, 61: 1619–1624

  15. 15

    Deng F Q, Luo Q, Mao X R, et al. Noise suppresses or expresses exponential growth. Syst Control Lett, 2008, 57: 262–270

  16. 16

    Liu L, Shen Y. Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition. Automatica, 2012, 48: 619–624

  17. 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

  18. 18

    Pavlović G, Janković S. Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay. J Comput Appl Math, 2012, 236: 1679–1690

  19. 19

    Wu F K, Hu S G. Stochastic suppression and stabilization of delay differential systems. Int J Robust Nonlinear Control, 2011, 21: 488–500

  20. 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

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Acknowledgements

This work was partially supported by National Natural Science Foundation of China (Grant No. 11571024), China Postdoctoral Science Foundation (Grant No. 2017M621588), Natural Science Foundation of Hebei Province of China (Grant No. A2015209229), Science and Technology Research Foundation of Higher Education Institutions of Hebei Province of China (Grant No. QN2017116), Grant From the Simons Foundation (Grant No. 429343, Renming Song), and Graduate Foundation of the North China University of Science and Technology (Grant No. K1603).

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Correspondence to Shoumei Li.

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Cite this article

Feng, L., Li, S., Song, R. et al. Suppression of explosion by polynomial noise for nonlinear differential systems. Sci. China Inf. Sci. 61, 70215 (2018). https://doi.org/10.1007/s11432-017-9340-4

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Keywords

  • explosion suppression
  • noise
  • non-autonomous differential system
  • growth with general rate
  • decay with general rate
  • general polynomial growth condition