Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

Tian Zhang; Chuanhou Gao

doi:10.3934/dcdsb.2021204

Article Contents

2022, Volume 27, Issue 7: 3725-3747. Doi: 10.3934/dcdsb.2021204

This issue Previous Article Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives Next Article Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

Tian Zhang^, and
Chuanhou Gao

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: Tian Zhang
^* Corresponding author: Tian Zhang

Received: December 31, 2020

Revised: May 31, 2021

Published: July 2022

This work was supported by the National Natural Science Foundation of China under Grant No. 12071428 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ20A010002

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Abstract

This paper focuses on the $ p $th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L$ \acute{e} $vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, $ H_\infty $ stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.

Keywords:

Neutral stochastic differential equations,
pantograph delay,
stability,
general decay,
Lévy noise.

Mathematics Subject Classification: 60H10, 93E15.

Citation:

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Figure 1. Poisson jump process

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Figure 2. State trajectories of two subsystems

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Figure 3. State trajectory of whole system

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Figure 4. Poisson jump process

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Figure 5. State trajectories of two subsystems

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Figure 6. State trajectory of whole system

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