Stabilization and destabilisation of non-autonomous stochastic nonlinear delay differential equations

Abstract

This paper focuses on a class of non-autonomous stochastic nonlinear delay differential equations, where the time delay functions are no longer required to be bounded or differentiable. The key aim is to study the stabilisation and destabilisation of the non-autonomous nonlinear stochastic delay differential equations by using the function ln ⁡ | x ( t ) | 2 , the improved LaSalle-type theorem, and non-negative semimartingale convergence theorem. In comparison with recent works on stabilisation for stochastic time-varying delay nonlinear systems (see, e.g. [Dong, H., & Mao, X. (2022). Advances in stabilization of highly nonlinear hybrid delay systems. Automatica, 136, 110086. https://doi.org/10.1016/j.automatica.2021.110086; Hu, J., Mao,W., & Mao, X. (2023). Advances in nonlinear hybrid stochastic differential delay equations: Existence, boundness and stability. Automatica, 147, 110682. https://doi.org/10.1016/j.automatica.2022.110682; Xu, H., & Mao, X. (2023). Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays. Nonlinear Analysis: Hybrid Systems, 50, 101413. https://doi.org/10.1016/j.nahs.2023.101413]), the proposed results exhibit significant improvements and can be applied to a broader class of non-autonomous equations.