A note on explosion suppression for nonlinear delay differential systems by polynomial noise

Abstract

Liu and Shen discussed the role of stochastic suppression on the explosive solution by a polynomial noise for a deterministic differential system satisfying a general polynomial growth condition. They further showed that the global solution of the corresponding perturbed system grows at most polynomially. However, the estimation of the asymptotic property of polynomial growth is rough, and we see the necessity to develop a more accurate estimation which is the main motivation of the present paper. As to the existence of time delays, we aim to discuss the stochastic roles of the polynomial noise for a deterministic delay differential system with the general polynomial growth condition. We show that a properly chosen polynomial stochastic noise not only can guarantee the existence and uniqueness of the global solution of the stochastically perturbed delay differential system, but also can make almost every sample path of the global solution grow at most with polynomial rate and even decay to the zero solution exponentially.

Keywords:

• Stochastic suppression
• stochastic stabilization
• stochastic noise
• delay differential system
• general polynomial growth condition

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project is partially supported by NSFC [No. 11571024], [No. 61673141]; China Postdoctoral Science Foundation [No. 2017M621588]; the Fok Ying Tung Education Foundation of China [No. 151004]; the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province [No. UNPYSCT-2016029]; Natural Science Foundation of Hebei Province of China [No. A2015209229]; Science and Technology Research Foundation of Higher Education Institutions of Hebei Province of China [No. QN2017116]; Returned Overseas Scholar Funding of Hebei Province [No. C2015005014]; Graduate Foundation of North China University of Science and Technology [No. K1603].