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doi: 10.3934/dcdss.2020370

On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks

Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA

* Corresponding author: Ivanka Stamova

Received  September 2019 Revised  January 2020 Published  May 2020

Fund Project: This paper is supported in part by the European Regional Development Fund through the Operational Programme "Science and Education for Smart Growth" under Contract UNITe No. BG05M2OP001–1.001–0004 (2018–2023)

In this paper, we introduce the notion of stability of sets for reaction-diffusion Cohen–Grossberg neural networks with time-varying delays. The Lyapunov–Razumikhin technique and a comparison principle are adapted to prove the new stability criteria. In addition, the obtained results are extended to the uncertain case, and the robust stability notion is also investigated. Examples are considered to demonstrate the effectiveness of our results.

Citation: Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020370
References:
[1]

C. Aouiti and E. A. Assali, Nonlinear Lipschitz measure and adaptive control for stability and synchronization in delayed inertial Cohen-Grossberg-type neural networks, Int. J. Adapt. Control, 33 (2019), 1457-1477.  doi: 10.1002/acs.3042.  Google Scholar

[2]

S. Baldwin and E. E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two manifolds, Proc. Amer. Math. Soc., 109 (1990), 823-828.  doi: 10.2307/2048225.  Google Scholar

[3]

M. Bohner, G. Tr. Stamov and I. M. Stamova, Almost periodic solutions of Cohen–Grossberg neural networks with time-varying delay and variable impulsive perturbations, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104952, 14pp. doi: 10.1016/j.cnsns.2019.104952.  Google Scholar

[4]

W. H. ChenL. Liu and X. Lu, Intermittent synchronization of reaction-diffusion neural networks with mixed delays via Razumikhin technique, Nonlinear Dyn., 87 (2017), 535-551.   Google Scholar

[5]

M. A. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., 13 (1983), 815-826.   Google Scholar

[6]

Q. Gan, Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3040-3049.  doi: 10.1016/j.cnsns.2011.11.012.  Google Scholar

[7]

Q. Gan, Exponential synchronization of generalized neural networks with mixed time-varying delays and reaction-diffusion terms via aperiodically intermittent control, Chaos, 27 (2017), 013113, 10pp. doi: 10.1063/1.4973976.  Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, 1$^st$ edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[9]

Y.-G. KaoJ.-F. GuoC.-H. Wang and X.-Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays, J. Franklin Inst., 349 (2012), 1972-1988.  doi: 10.1016/j.jfranklin.2012.04.005.  Google Scholar

[10]

Y. KaoC. Wang and L. Zhang, Delay-dependent robust exponential stability of impulsive Markovian jumping reaction-diffusion Cohen–Grossberg neural networks, Neural Process. Lett., 38 (2013), 321-346.   Google Scholar

[11]

H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Prentice-Hall, Upper Saddle River, NJ, 2002. Google Scholar

[12]

X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen–Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292-307.  doi: 10.1016/j.amc.2009.05.005.  Google Scholar

[13]

X. Li, Exponential stability of Cohen–Grossberg-type BAM neural networks with time-varying delays via impulsive control, Neurocomputing, 73 (2009), 525-530.   Google Scholar

[14]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.  Google Scholar

[15]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar

[16]

R. Li and J. Cao, Stability analysis of reaction-diffusion uncertain memristive neural networks with time-varying delays and leakage term, Appl. Math. Comput., 278 (2016), 54-69.  doi: 10.1016/j.amc.2016.01.016.  Google Scholar

[17]

R. LiJ. CaoA. Alsaedi and B. Ahmad, Passivity analysis of delayed reaction-diffusion Cohen–Grossberg neural networks via Hardy–Poincarè inequality, J. Franklin Inst., 354 (2017), 3021-3038.  doi: 10.1016/j.jfranklin.2017.02.028.  Google Scholar

[18]

L. Li and J. Jian, Exponential convergence and Lagrange stability for impulsive Cohen–Grossberg neural networks with time-varying delays, J. Comput. Appl. Math., 277 (2015), 23-35.  doi: 10.1016/j.cam.2014.08.029.  Google Scholar

[19]

B. LiuX. Liu and X. Liao, Robust stability of uncertain impulsive dynamical systems, J. Math. Anal. Appl., 290 (2004), 519-533.  doi: 10.1016/j.jmaa.2003.10.035.  Google Scholar

[20]

X. Lou and B. Cui, Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms, Chaos Solitons Fractals, 33 (2007), 653-662.  doi: 10.1016/j.chaos.2006.01.044.  Google Scholar

[21]

J. G. Lu, Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116-125.  doi: 10.1016/j.chaos.2007.05.002.  Google Scholar

[22]

F. M. G. Magpantay and A. R. Humphries, Generalized Lyapunov–Razumikhin techniques for scalar state-dependent delay differential equations, Discrete Continuous Dynam. Systems - S, 13 (2020), 85-104.  doi: 10.3934/dcdss.2020005.  Google Scholar

[23]

J. PanX. Liu and S. Zhong, Stability criteria for impulsive reaction-diffusion Cohen–Grossberg neural networks with time-varying delays, Math. Comput. Modelling, 51 (2010), 1037-1050.  doi: 10.1016/j.mcm.2009.12.004.  Google Scholar

[24]

J. Pan and S. Zhong, Dynamical behaviors of impulsive reaction-diffusion Cohen–Grossberg neural network with delays, Neurocomputing, 73 (2010), 1344-1351.   Google Scholar

[25]

A. PratapR. RajaJ. CaoC. P. Lim and O. Bagdasar, Stability and pinning synchronization analysis of fractional order delayed Cohen-Grossberg neural networks with discontinuous activations, Appl. Math. Comput., 359 (2019), 241-260.  doi: 10.1016/j.amc.2019.04.062.  Google Scholar

[26]

J. Qiu, Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing, 70 (2007), 1102-1108.   Google Scholar

[27]

R. RakkiyappanS. Dharani and Q. Zhu, Synchronization of reaction-diffusion neural networks with time-varying delays via stochastic sampled-data controller, Nonlinear Dyn., 79 (2015), 485-500.  doi: 10.1007/s11071-014-1681-x.  Google Scholar

[28]

R. Rao, Delay-dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen–Grossberg neural networks with partially known transition rates via Hardy–Poincarè inequality, Chin. Ann. Math., 35 (2014), 575-598.  doi: 10.1007/s11401-014-0839-7.  Google Scholar

[29]

F. R. Ruiz del Portal, Stable sets of planar homeomorphisms with translation pseudo-arcs, Discrete Continuous Dynam. Systems - S, 12 (2019), 2379-2390.  doi: 10.3934/dcdss.2019149.  Google Scholar

[30]

R. SkjetneT. I. Fossen and P. V. Kokotovic, Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory, Automatica, 41 (2005), 289-298.  doi: 10.1016/j.automatica.2004.10.006.  Google Scholar

[31]

I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations, 1$^st$ edition, Walter de Gruyter, Berlin, 2009. doi: 10.1515/9783110221824.  Google Scholar

[32]

I. M. Stamova and S. Simeonov, Delayed reaction-diffusion cellular neural networks of fractional order: Mittag–Leffler stability and synchronization, J. Comput. Nonlinear Dynam., 13 (2018), 011015 (7 pages). doi: 10.1115/1.4038290.  Google Scholar

[33]

I. M. Stamova and G. T. Stamov, On the stability of sets for delayed Kolmogorov-type systems, Proc. Amer. Math. Soc., 142 (2014), 591-601.  doi: 10.1090/S0002-9939-2013-12197-0.  Google Scholar

[34]

I. M. Stamova and G. T. Stamov, Applied Impulsive Mathematical Models, 1$^{st}$ edition, Springer (CMS Books in Mathematics), Cham, 2016. doi: 10.1007/978-3-319-28061-5.  Google Scholar

[35]

I. Stamova and G. Stamov, Mittag–Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers, Neural Netw., 96 (2017), 22-32.  doi: 10.1016/j.neunet.2017.08.009.  Google Scholar

[36]

C. Vidhya and P. Balasubramaniam, Robust stability of uncertain Markovian jumping stochastic Cohen–Grossberg type BAM neural networks with time-varying delays and reaction diffusion terms, Neural Parallel Sci. Comput., 19 (2011), 181-195.   Google Scholar

[37]

Y. WanJ. CaoG. Wen and W. Yu, Robust fixed–ime synchronization of delayed Cohen–Grossberg neural networks, Neural Netw., 73 (2016), 86-94.   Google Scholar

[38]

J.-L. WangH.-N. Wu and L. Guo, Stability analysis of reaction-diffusion Cohen–Grossberg neural networks under impulsive control, Neurocomputing, 106 (2013), 21-30.  doi: 10.1016/j.neucom.2012.11.006.  Google Scholar

[39]

Z. Wang and H. Zhang, Global asymptotic stability of reaction-diffusion Cohen–Grossberg neural networks with continuously distributed delays, IEEE Trans. Neral Netw., 21 (2010), 39-49.   Google Scholar

[40]

P. Yan and T. Lv, Periodicity of delayed reaction-diffusion high-order Cohen–Grossberg neural networks with Dirichlet boundary conditions, Rocky Mountain J. Math., 41 (2011), 949-970.  doi: 10.1216/RMJ-2011-41-3-949.  Google Scholar

[41]

X. YangJ. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive control, SIAM J. Control Opt., 51 (2013), 3486-3510.  doi: 10.1137/120897341.  Google Scholar

[42]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. - Hybri. Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar

[43]

T. Yoshizawa, Stability Theory by Liapunov's Second Method, 1$^{st}$ edition, The Mathematical Society of Japan, Tokyo, 1966.  Google Scholar

[44]

K. YuanJ. Cao and H. Li, Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays, IEEE Trans. Syst. Man Cybern., 36 (2006), 1356-1363.   Google Scholar

[45]

A. I. Zecevic and D. D. Siljak, Control of Complex Systems: Structural Constraints and Uncertainty, 1$^{st}$ edition, Springer, New York, 2010. Google Scholar

[46]

Q. Zhu and X. Li, Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks, Fuzzy Sets Syst., 203 (2012), 74-94.  doi: 10.1016/j.fss.2012.01.005.  Google Scholar

show all references

References:
[1]

C. Aouiti and E. A. Assali, Nonlinear Lipschitz measure and adaptive control for stability and synchronization in delayed inertial Cohen-Grossberg-type neural networks, Int. J. Adapt. Control, 33 (2019), 1457-1477.  doi: 10.1002/acs.3042.  Google Scholar

[2]

S. Baldwin and E. E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two manifolds, Proc. Amer. Math. Soc., 109 (1990), 823-828.  doi: 10.2307/2048225.  Google Scholar

[3]

M. Bohner, G. Tr. Stamov and I. M. Stamova, Almost periodic solutions of Cohen–Grossberg neural networks with time-varying delay and variable impulsive perturbations, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104952, 14pp. doi: 10.1016/j.cnsns.2019.104952.  Google Scholar

[4]

W. H. ChenL. Liu and X. Lu, Intermittent synchronization of reaction-diffusion neural networks with mixed delays via Razumikhin technique, Nonlinear Dyn., 87 (2017), 535-551.   Google Scholar

[5]

M. A. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., 13 (1983), 815-826.   Google Scholar

[6]

Q. Gan, Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3040-3049.  doi: 10.1016/j.cnsns.2011.11.012.  Google Scholar

[7]

Q. Gan, Exponential synchronization of generalized neural networks with mixed time-varying delays and reaction-diffusion terms via aperiodically intermittent control, Chaos, 27 (2017), 013113, 10pp. doi: 10.1063/1.4973976.  Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, 1$^st$ edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[9]

Y.-G. KaoJ.-F. GuoC.-H. Wang and X.-Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays, J. Franklin Inst., 349 (2012), 1972-1988.  doi: 10.1016/j.jfranklin.2012.04.005.  Google Scholar

[10]

Y. KaoC. Wang and L. Zhang, Delay-dependent robust exponential stability of impulsive Markovian jumping reaction-diffusion Cohen–Grossberg neural networks, Neural Process. Lett., 38 (2013), 321-346.   Google Scholar

[11]

H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Prentice-Hall, Upper Saddle River, NJ, 2002. Google Scholar

[12]

X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen–Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292-307.  doi: 10.1016/j.amc.2009.05.005.  Google Scholar

[13]

X. Li, Exponential stability of Cohen–Grossberg-type BAM neural networks with time-varying delays via impulsive control, Neurocomputing, 73 (2009), 525-530.   Google Scholar

[14]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.  Google Scholar

[15]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar

[16]

R. Li and J. Cao, Stability analysis of reaction-diffusion uncertain memristive neural networks with time-varying delays and leakage term, Appl. Math. Comput., 278 (2016), 54-69.  doi: 10.1016/j.amc.2016.01.016.  Google Scholar

[17]

R. LiJ. CaoA. Alsaedi and B. Ahmad, Passivity analysis of delayed reaction-diffusion Cohen–Grossberg neural networks via Hardy–Poincarè inequality, J. Franklin Inst., 354 (2017), 3021-3038.  doi: 10.1016/j.jfranklin.2017.02.028.  Google Scholar

[18]

L. Li and J. Jian, Exponential convergence and Lagrange stability for impulsive Cohen–Grossberg neural networks with time-varying delays, J. Comput. Appl. Math., 277 (2015), 23-35.  doi: 10.1016/j.cam.2014.08.029.  Google Scholar

[19]

B. LiuX. Liu and X. Liao, Robust stability of uncertain impulsive dynamical systems, J. Math. Anal. Appl., 290 (2004), 519-533.  doi: 10.1016/j.jmaa.2003.10.035.  Google Scholar

[20]

X. Lou and B. Cui, Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms, Chaos Solitons Fractals, 33 (2007), 653-662.  doi: 10.1016/j.chaos.2006.01.044.  Google Scholar

[21]

J. G. Lu, Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116-125.  doi: 10.1016/j.chaos.2007.05.002.  Google Scholar

[22]

F. M. G. Magpantay and A. R. Humphries, Generalized Lyapunov–Razumikhin techniques for scalar state-dependent delay differential equations, Discrete Continuous Dynam. Systems - S, 13 (2020), 85-104.  doi: 10.3934/dcdss.2020005.  Google Scholar

[23]

J. PanX. Liu and S. Zhong, Stability criteria for impulsive reaction-diffusion Cohen–Grossberg neural networks with time-varying delays, Math. Comput. Modelling, 51 (2010), 1037-1050.  doi: 10.1016/j.mcm.2009.12.004.  Google Scholar

[24]

J. Pan and S. Zhong, Dynamical behaviors of impulsive reaction-diffusion Cohen–Grossberg neural network with delays, Neurocomputing, 73 (2010), 1344-1351.   Google Scholar

[25]

A. PratapR. RajaJ. CaoC. P. Lim and O. Bagdasar, Stability and pinning synchronization analysis of fractional order delayed Cohen-Grossberg neural networks with discontinuous activations, Appl. Math. Comput., 359 (2019), 241-260.  doi: 10.1016/j.amc.2019.04.062.  Google Scholar

[26]

J. Qiu, Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing, 70 (2007), 1102-1108.   Google Scholar

[27]

R. RakkiyappanS. Dharani and Q. Zhu, Synchronization of reaction-diffusion neural networks with time-varying delays via stochastic sampled-data controller, Nonlinear Dyn., 79 (2015), 485-500.  doi: 10.1007/s11071-014-1681-x.  Google Scholar

[28]

R. Rao, Delay-dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen–Grossberg neural networks with partially known transition rates via Hardy–Poincarè inequality, Chin. Ann. Math., 35 (2014), 575-598.  doi: 10.1007/s11401-014-0839-7.  Google Scholar

[29]

F. R. Ruiz del Portal, Stable sets of planar homeomorphisms with translation pseudo-arcs, Discrete Continuous Dynam. Systems - S, 12 (2019), 2379-2390.  doi: 10.3934/dcdss.2019149.  Google Scholar

[30]

R. SkjetneT. I. Fossen and P. V. Kokotovic, Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory, Automatica, 41 (2005), 289-298.  doi: 10.1016/j.automatica.2004.10.006.  Google Scholar

[31]

I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations, 1$^st$ edition, Walter de Gruyter, Berlin, 2009. doi: 10.1515/9783110221824.  Google Scholar

[32]

I. M. Stamova and S. Simeonov, Delayed reaction-diffusion cellular neural networks of fractional order: Mittag–Leffler stability and synchronization, J. Comput. Nonlinear Dynam., 13 (2018), 011015 (7 pages). doi: 10.1115/1.4038290.  Google Scholar

[33]

I. M. Stamova and G. T. Stamov, On the stability of sets for delayed Kolmogorov-type systems, Proc. Amer. Math. Soc., 142 (2014), 591-601.  doi: 10.1090/S0002-9939-2013-12197-0.  Google Scholar

[34]

I. M. Stamova and G. T. Stamov, Applied Impulsive Mathematical Models, 1$^{st}$ edition, Springer (CMS Books in Mathematics), Cham, 2016. doi: 10.1007/978-3-319-28061-5.  Google Scholar

[35]

I. Stamova and G. Stamov, Mittag–Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers, Neural Netw., 96 (2017), 22-32.  doi: 10.1016/j.neunet.2017.08.009.  Google Scholar

[36]

C. Vidhya and P. Balasubramaniam, Robust stability of uncertain Markovian jumping stochastic Cohen–Grossberg type BAM neural networks with time-varying delays and reaction diffusion terms, Neural Parallel Sci. Comput., 19 (2011), 181-195.   Google Scholar

[37]

Y. WanJ. CaoG. Wen and W. Yu, Robust fixed–ime synchronization of delayed Cohen–Grossberg neural networks, Neural Netw., 73 (2016), 86-94.   Google Scholar

[38]

J.-L. WangH.-N. Wu and L. Guo, Stability analysis of reaction-diffusion Cohen–Grossberg neural networks under impulsive control, Neurocomputing, 106 (2013), 21-30.  doi: 10.1016/j.neucom.2012.11.006.  Google Scholar

[39]

Z. Wang and H. Zhang, Global asymptotic stability of reaction-diffusion Cohen–Grossberg neural networks with continuously distributed delays, IEEE Trans. Neral Netw., 21 (2010), 39-49.   Google Scholar

[40]

P. Yan and T. Lv, Periodicity of delayed reaction-diffusion high-order Cohen–Grossberg neural networks with Dirichlet boundary conditions, Rocky Mountain J. Math., 41 (2011), 949-970.  doi: 10.1216/RMJ-2011-41-3-949.  Google Scholar

[41]

X. YangJ. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive control, SIAM J. Control Opt., 51 (2013), 3486-3510.  doi: 10.1137/120897341.  Google Scholar

[42]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. - Hybri. Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar

[43]

T. Yoshizawa, Stability Theory by Liapunov's Second Method, 1$^{st}$ edition, The Mathematical Society of Japan, Tokyo, 1966.  Google Scholar

[44]

K. YuanJ. Cao and H. Li, Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays, IEEE Trans. Syst. Man Cybern., 36 (2006), 1356-1363.   Google Scholar

[45]

A. I. Zecevic and D. D. Siljak, Control of Complex Systems: Structural Constraints and Uncertainty, 1$^{st}$ edition, Springer, New York, 2010. Google Scholar

[46]

Q. Zhu and X. Li, Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks, Fuzzy Sets Syst., 203 (2012), 74-94.  doi: 10.1016/j.fss.2012.01.005.  Google Scholar

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