PPubMed13 Sep 2021Cited 1×

A New Criterion for Exponential Stability of a Class of Hopfield Neural Network with Time-Varying Delay Based on Gronwall's Inequality

Faming Guo, Ricai Luo, Xiaolan Qin, Yunfei Yi

Abstract

In this paper, we study the problem of exponential stability for the Hopfield neural network with time-varying delays. Different from the existing results, we establish new stability criteria by employing the method of variation of constants and Gronwall's integral inequality. Finally, we give several examples to show the effectiveness and applicability of the obtained criterion.

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A new exponential stability criterion for Hopfield neural networks with time‑varying delay is derived using variation of constants and Gronwall's inequality, avoiding unknown matrices.

What’s new

i.Derived new exponential stability criteria via variation of constants and Gronwall's inequality
ii.Criteria depend only on the system's coefficient matrices, eliminating the need for unknown Lyapunov matrices
iii.Provided four numerical examples illustrating the effectiveness and limits of the new criteria

How they did it

The authors apply the variation‑of‑constants formula to the Hopfield network equations and then bound the solution using Gronwall's integral inequality. This yields explicit sufficient conditions for global exponential stability expressed solely in terms of the coefficient matrices and delay parameters.

What they found

Example 1 (no delay): ω=2, ‖B‖=1.4, L‖B‖‑ω=‑0.6 ⇒ exponential stability
Example 2 (constant delay τ=0.1): (‖B‖+‖C‖)L exp(ωτ)‑ω≈‑0.0107 <0 ⇒ exponential stability
Example 3 (variable delay τ(t)=0.1 sin t+0.1): ((‖B‖+‖C‖)L/(1‑τ*) exp(ωτ))‑ω≈‑0.1767 <0 ⇒ exponential stability
Counter‑example: same system with larger ‖B‖,‖C‖ gives condition ≈33.31 >0 ⇒ not exponentially stable

Caveats

✕Conditions are only sufficient and may be conservative
✕Applicable only to Hopfield networks with Lipschitz continuous activation functions
✕Does not provide necessary conditions or address systems beyond the considered class

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