Energy-efficient intermittent control for cluster synchronization of discrete space–time stochastic retarded inertial neural networks

Dong Pan; Shaobin Rao; Dong Pan; Shaobin Rao

doi:10.3934/math.20261172

AIMS Mathematics

2026, Volume 11, Issue 9: 29529-29559. doi: 10.3934/math.20261172

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Research article

Energy-efficient intermittent control for cluster synchronization of discrete space–time stochastic retarded inertial neural networks

Dong Pan ^{1
,
,},
Shaobin Rao ²

1.
School of Basic Science, Guilin University of Technology at Nanning, Nanning 530001, Guangxi, China
2.
Applied Technology College of Soochow University, Suzhou 215325, China

Received: 01 July 2026 Revised: 20 August 2026 Accepted: 26 August 2026 Published: 14 September 2026
MSC : 34D06, 34K20, 92B20

This paper investigates mean-square cluster synchronization of discrete-time stochastic inertial neural networks with spatial diffusion, time-varying delays, and stochastic perturbations. A clustered drive–response framework is established via finite-difference approximation and Euler–Maruyama discretization. An intermittent boundary controller is proposed, acting only on boundary nodes for $ \beta $ steps within each period of length $ \theta $, thereby reducing the duty ratio from $ 100\% $ to $ \beta/\theta $. A six-component space–time discrete Lyapunov–Krasovskii functional is constructed, from which interval-dependent matrix-inequality conditions are derived for the control-active and control-inactive intervals. A periodwise contraction condition further links the two regimes and guarantees mean-square exponential cluster synchronization with an explicit convergence rate. Numerical simulations verify the effectiveness of the proposed scheme.
Citation: Dong Pan, Shaobin Rao. Energy-efficient intermittent control for cluster synchronization of discrete space–time stochastic retarded inertial neural networks[J]. AIMS Mathematics, 2026, 11(9): 29529-29559. doi: 10.3934/math.20261172

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Abstract

This paper investigates mean-square cluster synchronization of discrete-time stochastic inertial neural networks with spatial diffusion, time-varying delays, and stochastic perturbations. A clustered drive–response framework is established via finite-difference approximation and Euler–Maruyama discretization. An intermittent boundary controller is proposed, acting only on boundary nodes for $ \beta $ steps within each period of length $ \theta $, thereby reducing the duty ratio from $ 100\% $ to $ \beta/\theta $. A six-component space–time discrete Lyapunov–Krasovskii functional is constructed, from which interval-dependent matrix-inequality conditions are derived for the control-active and control-inactive intervals. A periodwise contraction condition further links the two regimes and guarantees mean-square exponential cluster synchronization with an explicit convergence rate. Numerical simulations verify the effectiveness of the proposed scheme.

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