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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

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  • ReceivedFeb 6, 2017
  • AcceptedApr 20, 2017
  • PublishedAug 30, 2017

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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