Synchronization in Probability of Reaction–Diffusion Neural Networks Under Stochastic Sampling Governed by Infinite Discrete Distributions
- 1Key Laboratory of Numerical Simulation of Sichuan Provincial Universities, School of Mathematics and Big Data, Neijiang Normal University, Neijiang 641100, China
Abstract
This paper generalizes stochastic sampled-data synchronization of reaction–diffusion neural networks (RDNNs) from sampling distributions with finite support to infinite discrete distributions, whose sampling intervals are drawn from a countably infinite set {qi}i=1∞ with probabilities {αi}i=1∞. Four regularity conditions are identified—finite interval mean, summability of the weighted absolute decay parameters, strict positivity of the probability-averaged decay rate, and boundedness above of the decay parameters, which makes the Lyapunov–Krasovskii functional (LKF) mode-independent in its delay weight and hence continuous across sampling instants. A mode-dependent LKF is constructed, the weak-law step is replaced by the strong law of large numbers combined with a Cesàro-type argument, and the infinite family of design linear matrix inequalities (LMIs) is handled by a truncation-with-tail-verification procedure, for which a rigorous certificate is given: when the tail decay parameters are non-positive, the tail conditions are affine and are certified by corner-point feasibility checks with shared variables; otherwise, a convergent-tail argument applies. Almost-sure exponential synchronization with an explicit rate is established, which implies synchronization in probability, and controller design conditions are derived in terms of LMIs; three examples (two neural networks and one reaction–diffusion neural network) corroborate the design through computed gains, verified tail conditions, stochastic closed-loop simulations, an N-sensitivity analysis, and a comparison with a fixed-period controller. The proposed framework removes the finite-support restriction on the sampling distribution and yields an almost-sure, rate-explicit synchronization guarantee with a numerically verifiable LMI design.
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BibTeX
@article{zeng2026synchronization,
title = {{Synchronization in Probability of Reaction–Diffusion Neural Networks Under Stochastic Sampling Governed by Infinite Discrete Distributions}},
author = {Deqiang Zeng},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203665},
url = {https://www.mdpi.com/2227-7390/14/20/3665},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Synchronization in Probability of Reaction–Diffusion Neural Networks Under Stochastic Sampling Governed by Infinite Discrete Distributions
AU - Deqiang Zeng
PY - 2026
DA - 2026/10/10
JO - Mathematics
DO - 10.3390/math14203665
UR - https://www.mdpi.com/2227-7390/14/20/3665
AB - This paper generalizes stochastic sampled-data synchronization of reaction–diffusion neural networks (RDNNs) from sampling distributions with finite support to infinite discrete distributions, whose sampling intervals are drawn from a countably infinite set {qi}i=1∞ with probabilities {αi}i=1∞. Four regularity conditions are identified—finite interval mean, summability of the weighted absolute decay parameters, strict positivity of the probability-averaged decay rate, and boundedness above of the decay parameters, which makes the Lyapunov–Krasovskii functional (LKF) mode-independent in its delay weight and hence continuous across sampling instants. A mode-dependent LKF is constructed, the weak-law step is replaced by the strong law of large numbers combined with a Cesàro-type argument, and the infinite family of design linear matrix inequalities (LMIs) is handled by a truncation-with-tail-verification procedure, for which a rigorous certificate is given: when the tail decay parameters are non-positive, the tail conditions are affine and are certified by corner-point feasibility checks with shared variables; otherwise, a convergent-tail argument applies. Almost-sure exponential synchronization with an explicit rate is established, which implies synchronization in probability, and controller design conditions are derived in terms of LMIs; three examples (two neural networks and one reaction–diffusion neural network) corroborate the design through computed gains, verified tail conditions, stochastic closed-loop simulations, an N-sensitivity analysis, and a comparison with a fixed-period controller. The proposed framework removes the finite-support restriction on the sampling distribution and yields an almost-sure, rate-explicit synchronization guarantee with a numerically verifiable LMI design.
SN - 2227-7390
ER -