Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

TL;DR

This paper establishes new multi-condition criteria for the stability in probability of nonlinear stochastic delay differential equations with time-varying coefficients, extending existing results to more complex systems.

Contribution

It introduces a set of sufficient conditions that ensure stability in probability for a broad class of nonlinear stochastic delay differential equations, including multiple delays and non-autonomous coefficients.

Findings

Derived new stability conditions for nonlinear stochastic delay equations.

Showed that linear stability conditions imply nonlinear stability in probability.

Validated results through numerical simulations and illustrative figures.

Abstract

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.