Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps

Wensheng Yin; Jinde Cao

doi:10.3934/dcdsb.2020109

Article Contents

2020, Volume 25, Issue 11: 4493-4513. Doi: 10.3934/dcdsb.2020109

This issue Previous Article Spreading speeds for a class of non-local convolution differential equation Next Article A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps

Wensheng Yin^1, and
Jinde Cao^1,2, ,

1.
School of Mathematics, Southeast University, Nanjing 211189, China
2.
Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 211189, China

^* Corresponding author: Jinde Cao
^* Corresponding author: Jinde Cao

Received: December 31, 2018

Revised: October 31, 2019

Early access: March 2020

Published: November 2020

This work was jointly supported by the Key Project of National Science Foundation of China under Grant No. 61833005

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Abstract

The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some sufficient criteria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, sufficient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period $ T $ and noise width $ \theta $. As for applications, the well-known Lorenz chaotic systems and nonlinear Liénard equation with jumps are discussed. Finally, two simulation examples demonstrating the effectiveness of the results are provided.

Keywords:

Almost sure exponential stabilization,
destabilization,
stochastic systems,
periodically intermittent stochastic perturbation,
Poisson jumps.

Mathematics Subject Classification: Primary: 60H10, 93E03; Secondary: 93D15.

Citation:

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Figure 1. The state $ x(t) $ of the system (29)

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Figure 2. The state $ x(t) $ of system (30)

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Figure 5. The exponent dynamical behaviors of state $ x(t) $ of the system (34)

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Figure 3. The state $ x(t) $ of the system (32)

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Figure 4. The state $ x(t) $ of system (33)

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Figure 6. The state $ x(t) $ of system (34)

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