Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses
Abstract
:1. Introduction
- (1)
- Different from the deterministic distributed delayed impulses in the literature [26,27,28,29,30,31], in this paper, the intensities of distributed delayed impulses are supposed to be random. Two types of stochastic impulses including stochastic distributed delayed impulses with independent property and Markov property have been explored, respectively.
- (2)
- Based on the average impulsive interval method, total probability formula and ergodic theory, two novel impulsive Halanay differential inequalities are established, which generalize the findings in the literature [34,35] since time-varying delays, distributed delays and stochastic distributed delayed impulses are introduced simultaneously. Parameter c can be arbitrarily chosen. In view of invariant distribution theory, the stochastic impulses with Markov property are tackled.
- (3)
- By utilizing the established inequalities and graph theory, some criteria for exponential synchronization of CNNs with stochastic distributed delayed impulses are derived. In [36], impulses can only be regarded as outer disturbances for coupled inertial NNs with hybrid delays. Compared with the work [36], in this paper, impulses may also be viewed as outer perturbations or stabilizing sources, and the case of stochastic impulses with Markov property is also discussed.
2. Preliminaries
3. Main Results
- (i)
- , where , , , , , and .
- (ii)
- , where , , , , and is the unique solution of .
- (i)
- , where , , , , , and ;
- (ii)
- , where , , , , , , , is a sufficient small positive constant and is the unique solution of ;
4. Examples
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Zhang, G.; Song, Y.; Liu, X. Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses. Mathematics 2024, 12, 1995. https://doi.org/10.3390/math12131995
Zhang G, Song Y, Liu X. Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses. Mathematics. 2024; 12(13):1995. https://doi.org/10.3390/math12131995
Chicago/Turabian StyleZhang, Gang, Yinfang Song, and Xiaoyou Liu. 2024. "Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses" Mathematics 12, no. 13: 1995. https://doi.org/10.3390/math12131995
APA StyleZhang, G., Song, Y., & Liu, X. (2024). Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses. Mathematics, 12(13), 1995. https://doi.org/10.3390/math12131995