Boundedness and stability of nonlinear hybrid stochastic differential delay equation disturbed by Lévy noise

Abstract

This study focuses on the boundedness and stability of the nonlinear hybrid stochastic differential delay equation (NHSDDE) disturbed by Lévy noise (LN). Different from the previous systems with Markovian switching, not only different parameters under different modes, but also different structures under different modes are considered, where a novel Lyapunov function is constructed to handle the impacts of different structures under different modes. Without the linear growth condition, the existence of the global solution is proved for the NHSDDE with LN. Then, the M‐matrix technique is applied to build some reasonable theorems upon the vth moment asymptotic boundedness, the vth moment exponential stability and the almost surely exponential stability for the NHSDDE driven by LN. An approach to solving the admissible upper bound of the time‐delay derivative is given for guaranteeing the asymptotic boundedness and the exponential stability of the global solution. Finally, two numerical examples and homologous simulated outcomes are provided to indicate the usefulness of the presented theoretical findings.

Stabilization of hybrid stochastic differential equations with multiple delays via intermittent control

“…As we all know, delays may cause oscillations and instability of systems. Therefore, it has practical significance to study the stability of hybrid SDEs with time delays (see [12][13][14][15]). In addition, SDEs not only rely on a single time delay, but may also be affected by multiple delays.…”