Quantized $ \mathcal{H}_\infty $ synchronization for Hopfield networks subject to time-variable delay

Tianran Bu; Tianran Bu

doi:10.3934/nhm.2025060

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1392-1410. doi: 10.3934/nhm.2025060

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Quantized $ \mathcal{H}_\infty $ synchronization for Hopfield networks subject to time-variable delay

Tianran Bu ^,

School of Information and Artificial Intelligence, Anhui Business College, Wuhu 241002, China

Received: 27 October 2025 Revised: 08 December 2025 Accepted: 10 December 2025 Published: 15 December 2025

This paper is dedicated to the study of $ \mathcal{H}_\infty $ synchronization for Hopfield networks (HNs) with time-variable delay via quantized control. The delay function is assumed to be continuous and bounded, yet not necessarily differentiable. A delay-dependent sufficient condition is derived using a linear combination of Lyapunov functional candidates along with two matrix inequalities to ensure the $ \mathcal{H}_\infty $ synchronization of drive-response HNs. Based on this, an alternative sufficient condition characterized by reduced nonlinearities is further established by employing decoupling techniques. With fixed bounds on the time delay and the $ \mathcal{H}_\infty $ disturbance attenuation level, an algorithm is then proposed to compute the minimum required gain for the quantized controller. Finally, a numerical example demonstrates the effectiveness of the proposed $ \mathcal{H}_\infty $ synchronization results.
- Hopfield network,
- quantized control,
- $ \mathcal{H}_\infty $ synchronization,
- time-variable delay
Citation: Tianran Bu. Quantized $ \mathcal{H}_\infty $ synchronization for Hopfield networks subject to time-variable delay[J]. Networks and Heterogeneous Media, 2025, 20(4): 1392-1410. doi: 10.3934/nhm.2025060

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Abstract

This paper is dedicated to the study of $ \mathcal{H}_\infty $ synchronization for Hopfield networks (HNs) with time-variable delay via quantized control. The delay function is assumed to be continuous and bounded, yet not necessarily differentiable. A delay-dependent sufficient condition is derived using a linear combination of Lyapunov functional candidates along with two matrix inequalities to ensure the $ \mathcal{H}_\infty $ synchronization of drive-response HNs. Based on this, an alternative sufficient condition characterized by reduced nonlinearities is further established by employing decoupling techniques. With fixed bounds on the time delay and the $ \mathcal{H}_\infty $ disturbance attenuation level, an algorithm is then proposed to compute the minimum required gain for the quantized controller. Finally, a numerical example demonstrates the effectiveness of the proposed $ \mathcal{H}_\infty $ synchronization results.

References

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