Mean-square exponential synchronization of coupled neural networks with non-Markovian noise via adaptive control

Zixin Zhu; Weihua Wan; Yuming Feng; Xiangguang Dai; Zixin Zhu; Weihua Wan; Yuming Feng; Xiangguang Dai

doi:10.3934/math.20261153

AIMS Mathematics

2026, Volume 11, Issue 9: 28991-29021. doi: 10.3934/math.20261153

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Mean-square exponential synchronization of coupled neural networks with non-Markovian noise via adaptive control

1.
School of Mathematics and Statistics, Chongqing Sanxia University of Science and Technology, Wanzhou, Chongqing 404100, China
2.
School of Intelligent Technology, Chongqing Preschool Education College, Wanzhou, Chongqing 404047, China
3.
School of Computer Science and Engineering of Science and Engineering, Chongqing Sanxia University of Science and Technology, Wanzhou, Chongqing 404100, China

Received: 13 May 2026 Revised: 10 August 2026 Accepted: 19 August 2026 Published: 09 September 2026
MSC : 60H10, 93C10, 93D20

In this paper, we investigated the synchronization problem of neural networks under non-Markovian disturbances. By incorporating fractional Brownian motion (fBm) and Markovian switching, the proposed model could characterize long-memory effects in stochastic systems. A hybrid adaptive control strategy, including gain adjustment, memory-kernel feedback, and distributed compensation, was developed to achieve synchronization. Sufficient conditions for mean-square exponential stability were derived using a Lyapunov functional approach. Numerical results showed that the proposed controller achieved a settling time of $ 0.91s $, compared with $ 3.75s $, $ 4.55s $, and $ 7.99s $ for the the Linear Feedback Controller (LFC), Proportional-Derivative (PD), and Proportional-Integral-Derivative (PID) controllers, respectively, while achieving smaller Integral Square Error (ISE), Integral Absolute Error (IAE), and Integral Time Absolute Error (ITAE) values.
- non-Markovian dynamics,
- dynamic adaptation,
- mean square exponential stability,
- fractional Brownian motion
Citation: Zixin Zhu, Weihua Wan, Yuming Feng, Xiangguang Dai. Mean-square exponential synchronization of coupled neural networks with non-Markovian noise via adaptive control[J]. AIMS Mathematics, 2026, 11(9): 28991-29021. doi: 10.3934/math.20261153

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Abstract

In this paper, we investigated the synchronization problem of neural networks under non-Markovian disturbances. By incorporating fractional Brownian motion (fBm) and Markovian switching, the proposed model could characterize long-memory effects in stochastic systems. A hybrid adaptive control strategy, including gain adjustment, memory-kernel feedback, and distributed compensation, was developed to achieve synchronization. Sufficient conditions for mean-square exponential stability were derived using a Lyapunov functional approach. Numerical results showed that the proposed controller achieved a settling time of $ 0.91s $, compared with $ 3.75s $, $ 4.55s $, and $ 7.99s $ for the the Linear Feedback Controller (LFC), Proportional-Derivative (PD), and Proportional-Integral-Derivative (PID) controllers, respectively, while achieving smaller Integral Square Error (ISE), Integral Absolute Error (IAE), and Integral Time Absolute Error (ITAE) values.

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