Globally exponential stability of delayed impulsive functional differential systems with impulse time windows

In the practical application of impulsive differential systems, impulse does not always occur at the fixed-time point; it may occur in a little range of time. Namely, impulse occurs in a time window, which is more general and more nearing to reality than those fixed-time impulses. Therefore, it is necessary to investigate the dynamical behaviors of impulsive differential systems with impulse time windows. In this paper, the exponential stability of these systems is researched. By means of Lyapunov functions, Razumikhin technique and other analysis methods, several novel exponential stability criteria for delayed impulsive functional differential equations with impulse time windows are obtained, which are different from the previously published results for fixed-time impulses. What is more, based on the analysis of this paper, it is worth noting that choosing an efficient impulse time window may be easier and more effective than choosing fixed-time impulsive sequences. Finally, three examples and their simulations are provided to illustrate the effectiveness of our results.

[1]  Jin Zhou,et al.  Synchronization in complex delayed dynamical networks with impulsive effects , 2007 .

[2]  Zhigang Zeng,et al.  Exponential Stabilization of Memristive Neural Networks With Time Delays , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[3]  Wei Xing Zheng,et al.  Exponential stability of nonlinear time-delay systems with delayed impulse effects , 2011, Autom..

[4]  Chuandong Li,et al.  Robust stability of stochastic fuzzy delayed neural networks with impulsive time window , 2015, Neural Networks.

[5]  Xinzhi Liu,et al.  Exponential stability for impulsive delay differential equations by Razumikhin method , 2005 .

[6]  Xilin Fu,et al.  Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems , 2009 .

[7]  Jinde Cao,et al.  Synchronization of delayed complex dynamical networks with impulsive and stochastic effects , 2011 .

[8]  Zhigang Zeng,et al.  Exponential passivity of memristive neural networks with time delays , 2014, Neural Networks.

[9]  Jianhua Shen,et al.  Asymptotic behavior of solutions of impulsive neutral differential equations , 1999 .

[10]  Xing Xin,et al.  Exponential stability of delayed and impulsive cellular neural networks with partially Lipschitz continuous activation functions , 2012, Neural Networks.

[11]  L. Berezansky,et al.  Exponential Stability of Linear Delay Impulsive Differential Equations , 1993 .

[12]  Feiqi Deng,et al.  Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses , 2014 .

[13]  Yang Liu,et al.  New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays , 2012, Math. Comput. Model..

[14]  Xinzhi Liu,et al.  Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method , 2007, Appl. Math. Lett..

[15]  X. Liao,et al.  Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols , 2012, Second-Order Consensus of Continuous-Time Multi-Agent Systems.

[16]  Gai Sun,et al.  Exponential stability of impulsive discrete-time stochastic BAM neural networks with time-varying delay , 2014, Neurocomputing.

[17]  Xinzhi Liu,et al.  Stability Criteria for Impulsive Systems With Time Delay and Unstable System Matrices , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[18]  Jinde Cao,et al.  Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations , 2014, Appl. Math. Comput..

[19]  Yi Zhang,et al.  Exponential Stability of Singularly Perturbed Systems with Time Delay , 2003 .

[20]  Huamin Wang,et al.  Stability of impulsive delayed linear differential systems with delayed impulses , 2015, J. Frankl. Inst..

[21]  Wenwu Yu,et al.  Impulsive synchronization schemes of stochastic complex networks with switching topology: Average time approach , 2014, Neural Networks.

[22]  Jin Zhou,et al.  Global exponential stability of impulsive differential equations with any time delays , 2010, Appl. Math. Lett..

[23]  Tingwen Huang,et al.  Accelerated consensus to accurate average in multi-agent networks via state prediction , 2013 .

[24]  Tao Yang,et al.  Impulsive control , 1999, IEEE Trans. Autom. Control..

[25]  Guanrong Chen,et al.  On delayed impulsive Hopfield neural networks , 1999, Neural Networks.

[26]  Xinzhi Liu,et al.  Application of Impulsive Synchronization to Communication Security , 2003 .

[27]  Leonid Berezansky,et al.  Exponential stability of some scalar impulsive delay differential equation , 1998 .

[28]  Jinde Cao,et al.  Synchronization of Coupled Reaction-Diffusion Neural Networks with Time-Varying Delays via Pinning-Impulsive Controller , 2013, SIAM J. Control. Optim..

[29]  José J. Oliveira,et al.  Stability results for impulsive functional differential equations with infinite delay , 2012 .

[30]  Donal O'Regan,et al.  Stability analysis of generalized impulsive functional differential equations , 2012, Math. Comput. Model..

[31]  Xinzhi Liu,et al.  The method of Lyapunov functionals and exponential stability of impulsive systems with time delay , 2007 .

[32]  Jinde Cao,et al.  Exponential stability of impulsive stochastic functional differential equations , 2011 .

[33]  Jinde Cao,et al.  Anti-periodic solution for delayed cellular neural networks with impulsive effects , 2011 .

[34]  Wu-Hua Chen,et al.  Exponential stability of a class of nonlinear singularly perturbed systems with delayed impulses , 2013, J. Frankl. Inst..

[35]  Tingwen Huang,et al.  Impulsive control and synchronization of nonlinear system with impulse time window , 2014 .

[36]  Zhiguo Luo,et al.  Stability of impulsive functional differential equations via the Liapunov functional , 2009, Appl. Math. Lett..

[37]  Yu Zhang,et al.  Stability of impulsive functional differential equations , 2008 .