Noise-to-state and delay-dependent stability of highly nonlinear hybrid stochastic differential equation with multiple delays

Abstract

This study is intended to discuss the noise-to-state stability (NSS) and the delay-dependent stability (DDS) of the highly nonlinear hybrid stochastic differential equation (HNHSDE) with multiple delays. Distinct from the white noise considered in the existing literature, we introduce the coloured noise characterised by a finite second-order process into the HNHSDE. Under some mild assumption conditions, we demonstrate that the HNHSDE exists a unique global solution (GS). Moreover, some criteria regarding the NSS in the qth moment of the GS are established. Furthermore, several theorems are provided for guaranteeing the pth moment asymptotic stability and the almost surely asymptotic stability of the GS with the aid of the delay-dependent Lyapunov functional and stochastic analysis theory. At last, two illustrative examples are given to show the feasibility of our developed theory findings.

Keywords:

• Highly nonlinear hybrid stochastic differential equation
• noise-to-state stability in the qth moment
• pth moment asymptotic stability
• almost surely asymptotic stability
• multiple delays

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 12071102, 12171124]; Natural Science Foundation of Heilongjiang Province of China [grant number ZD2022F003].