# SCIENCE CHINA Information Sciences, Volume 61, Issue 1: 012203(2018) https://doi.org/10.1007/s11432-017-9097-1

## Impulsive control of unstable neural networks with unbounded time-varying delays • ReceivedFeb 6, 2017
• AcceptedApr 20, 2017
• PublishedAug 30, 2017
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### Abstract

This paper considers the impulsive control of unstable neural networks with unbounded time-varying delays, where the time delays to be addressed include the unbounded discrete time-varying delay and unbounded distributed time-varying delay.By employing impulsive control theory and some analysis techniques, several sufficient conditions ensuring $\mu$-stability, including uniform stability, (global) asymptotical stability, and (global) exponential stability, are derived. It is shown that an unstable delay neural network, especially for the case of unbounded time-varying delays, can be stabilized and has $\mu$-stability via proper impulsive control strategies. Three numerical examples and their simulations are presented to demonstrate the effectivenessof the control strategy.

### Acknowledgment

This work was jointly supported by National Natural Science Foundation of China (Grant Nos. 11301308, 61673247, 61273233), Outstanding Youth Foundation of Shandong Province (Grant Nos. ZR20170- 2100145, ZR2016J L024), and Natural Sciences and Engineering Research Council of Canada.

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• Figure 1

(Color online) State trajectories $x=(x_1,x_2)^{\rm~T}$ of model (3) with different delays $\tau$, where red lines denote the trajectories of $x_1$, blue lines denote the trajectories of $x_2$. (a) $\tau$=0; (b) $\tau$= 3; (c) $\tau$=8; (d) $\tau$=0.5$t$.

• Figure 2

(Color online) State trajectories and phase plots of system (sect. 4) in Example 1. (a) State trajectories of $2D$ (sect. 4) without impulses; (b) phase plots of $2D$ (sect. 4) without impulses; (c) state trajectories of $2D$ (sect. 4) with impulsive input (28); (d) phase plots of $2D$ (sect. 4) with input (28); (e) state trajectories of $2D$ (sect. 4) with input (29).

• Figure 3

(Color online) State trajectories of system (sect. 4) in Example 2. (a) State trajectories of $3D$ (sect. 4) without impulses; (b) state trajectories of $3D$ (sect. 4) with impulsive input (30) and $t_k=0.035k,~k\in~\mathbb{Z}_+$; (c) state trajectories of $3D$ (sect. 4) with $t_k=0.065k$ and $\Gamma_k=-1.9129I$.

• Figure 4

(Color online) State trajectories of system (31) in Example 3. (a) State trajectories of (31) without impulses; (b) state trajectories of (31) with impulsive input (32) in which $t_k=0.33k$ and $\Gamma=-1.4714,~k\in~\mathbb{Z}_+$; (c) state trajectories of (31) with impulsive input (32) in which $t_k=0.5k$ and $\Gamma=-1.4714,~k\in~\mathbb{Z}_+$.

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