Existence and Exponential Stability of Almost Periodic Solution for Cohen-Grossberg SICNNs with Impulses
In this paper, based on the estimation of the Cauchy matrix of linear impulsive differential equations, by using Banach fixed point theorem and Gronwall-Bellman-s inequality, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays and impulses. An example is given to illustrate the main results.
[1] A. Bouzerdoum, R.B. Pinter, Analysis and analog implementation of
directionally sensitive shunting inhibitory cellular neural networks, Visual
Information Processing: From Neurons to Chips, vol. 1473, SPIE, 1991,
pp. 29-38.
[2] A. Bouzerdoum, R.B. Pinter, Nonlinear lateral inhibition applied to
motion detection in the fly visual system, in: R.B. Pinter, B. Nabet (Eds.),
Nonlinear Vision, CRC Press, Boca Raton, FL, 1992, pp. 423-450.
[3] A. Bouzerdoum, R.B. Pinter, Shunting inhibitory cellular neural networks:
derivation and stability analysis, IEEE Trans. Circuits and Systems 1-
Fundamental Theory and Applications 40 (1993) 215-221.
[4] M. Cai, W. Xiong, Almost periodic solutions for shunting inhibitory
cellular neural networks without global Lipschitz and bounded activation
functions, Phys. Lett. A 362 (2007) 417-423.
[5] M. Cai, H. Zhang, Z. Yuan, Positive almost periodic solutions for
shunting inhibitory cellular neural networks with time-varying delays,
Math. Comput. Simul. 78 (2008) 548-558.
[6] A. Chen, J. Cao, L. Huang, Almost periodic solution of shunting
inhibitory CNNs with delays, Phys. Lett. A 298 (2002) 161-170.
[7] X. Huang, J. Cao, Almost periodic solutions of inhibitory cellular neural
networks with time-vary delays, Phys. Lett. A 314 (2003) 222-231.
[8] Y.K. Li, C. Liu, L. Zhu, Global exponential stability of periodic solution
of shunting inhibitory CNNs with delays, Phys. Lett.A 337 (2005) 46-54.
[9] B. Liu, L. Huang, Existence and stability of almost periodic solutions for
shunting inhibitory cellular neural networks with continuously distributed
delays, Phys. Lett. A 349 (2006) 177-186.
[10] B. Liu, L. Huang, Existence and stability of almost periodic solutions for
shunting inhibitory cellular neural networks with continuously distributed
delays, Phys. Lett. A 349 (2006) 177-186.
[11] S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial
neural networks with distributed delays and large impulses, Nonlinear
Anal. 9 (2008) 872-888.
[12] L. Zhou, W. Li, Y. Chen, Suppressing the periodic impulse in partial
discharge detection based on chaotic control, Automat. Electric Power
Syst. 28 (2004) 90-94.
[13] V. Nagesh, Automatic detection and elimination of periodic pulse shaped
interference in partial discharge measurements, IEE Proc. Meas. Technol.
141 (1994) 335-339.
[14] C. Bai, Stability analysis of Cohen-Crossberg BAM neural networks
with delays and impulses, Chaos, Solitons & Fractals 35 (2008) 263-267.
[15] Z. Yang, D. Xu, Existence and exponential stability of periodic solution
for impulsive delay differential equations and applications, Nonlinear
Anal. 64 (2006) 130-145.
[16] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic
Solutions and Applications, Longman Scientific and Technical Group
Limited, New York, 1993.
[17] Y. Li, L. Lu, Global exponential stability and existence of periodic
solution of Hopfield-type neural networks with impulses Phys. Lett. A
333 (2004) 62-71.
[18] Y. Li, Global exponential stability of BAM neural networks with delays
and impulses Chaos, Solitons & Fractals, 24 (2005) 279-285.
[19] A.M. Fink, Almost Periodic Differential Equations, Springer, Berlin,
1974.
[20] D. Cheban, C. Mammana, Invariant manifolds, global attractors and
almost periodic solutions of nonautonomous difference equations, Nonlinear
Anal. 56(4) (2004) 465-484.
[21] A.M. Samoilenko, N.A. Perestyuk, Differential Equations with Impulse
Effect, World Scientific, Singapore, 1995.
[22] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive
Differential Equations, World Scientific, Singapore, New Jersey, London,
1989.
[23] G.T. Stamov, On the existence of almost periodic solutions for the
impulsive Lasota-Wazewska model, Appl. Math. Lett. 22 (2009) 516-520.
[24] Y. Li, T. Zhang, Z. Xing, The existence of nonzero almost periodic solution
for Cohen-Grossberg neural networks with continuously distributed
delays and impulses, Neurocomputing 73 (2010) 3105-3113.
[1] A. Bouzerdoum, R.B. Pinter, Analysis and analog implementation of
directionally sensitive shunting inhibitory cellular neural networks, Visual
Information Processing: From Neurons to Chips, vol. 1473, SPIE, 1991,
pp. 29-38.
[2] A. Bouzerdoum, R.B. Pinter, Nonlinear lateral inhibition applied to
motion detection in the fly visual system, in: R.B. Pinter, B. Nabet (Eds.),
Nonlinear Vision, CRC Press, Boca Raton, FL, 1992, pp. 423-450.
[3] A. Bouzerdoum, R.B. Pinter, Shunting inhibitory cellular neural networks:
derivation and stability analysis, IEEE Trans. Circuits and Systems 1-
Fundamental Theory and Applications 40 (1993) 215-221.
[4] M. Cai, W. Xiong, Almost periodic solutions for shunting inhibitory
cellular neural networks without global Lipschitz and bounded activation
functions, Phys. Lett. A 362 (2007) 417-423.
[5] M. Cai, H. Zhang, Z. Yuan, Positive almost periodic solutions for
shunting inhibitory cellular neural networks with time-varying delays,
Math. Comput. Simul. 78 (2008) 548-558.
[6] A. Chen, J. Cao, L. Huang, Almost periodic solution of shunting
inhibitory CNNs with delays, Phys. Lett. A 298 (2002) 161-170.
[7] X. Huang, J. Cao, Almost periodic solutions of inhibitory cellular neural
networks with time-vary delays, Phys. Lett. A 314 (2003) 222-231.
[8] Y.K. Li, C. Liu, L. Zhu, Global exponential stability of periodic solution
of shunting inhibitory CNNs with delays, Phys. Lett.A 337 (2005) 46-54.
[9] B. Liu, L. Huang, Existence and stability of almost periodic solutions for
shunting inhibitory cellular neural networks with continuously distributed
delays, Phys. Lett. A 349 (2006) 177-186.
[10] B. Liu, L. Huang, Existence and stability of almost periodic solutions for
shunting inhibitory cellular neural networks with continuously distributed
delays, Phys. Lett. A 349 (2006) 177-186.
[11] S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial
neural networks with distributed delays and large impulses, Nonlinear
Anal. 9 (2008) 872-888.
[12] L. Zhou, W. Li, Y. Chen, Suppressing the periodic impulse in partial
discharge detection based on chaotic control, Automat. Electric Power
Syst. 28 (2004) 90-94.
[13] V. Nagesh, Automatic detection and elimination of periodic pulse shaped
interference in partial discharge measurements, IEE Proc. Meas. Technol.
141 (1994) 335-339.
[14] C. Bai, Stability analysis of Cohen-Crossberg BAM neural networks
with delays and impulses, Chaos, Solitons & Fractals 35 (2008) 263-267.
[15] Z. Yang, D. Xu, Existence and exponential stability of periodic solution
for impulsive delay differential equations and applications, Nonlinear
Anal. 64 (2006) 130-145.
[16] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic
Solutions and Applications, Longman Scientific and Technical Group
Limited, New York, 1993.
[17] Y. Li, L. Lu, Global exponential stability and existence of periodic
solution of Hopfield-type neural networks with impulses Phys. Lett. A
333 (2004) 62-71.
[18] Y. Li, Global exponential stability of BAM neural networks with delays
and impulses Chaos, Solitons & Fractals, 24 (2005) 279-285.
[19] A.M. Fink, Almost Periodic Differential Equations, Springer, Berlin,
1974.
[20] D. Cheban, C. Mammana, Invariant manifolds, global attractors and
almost periodic solutions of nonautonomous difference equations, Nonlinear
Anal. 56(4) (2004) 465-484.
[21] A.M. Samoilenko, N.A. Perestyuk, Differential Equations with Impulse
Effect, World Scientific, Singapore, 1995.
[22] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive
Differential Equations, World Scientific, Singapore, New Jersey, London,
1989.
[23] G.T. Stamov, On the existence of almost periodic solutions for the
impulsive Lasota-Wazewska model, Appl. Math. Lett. 22 (2009) 516-520.
[24] Y. Li, T. Zhang, Z. Xing, The existence of nonzero almost periodic solution
for Cohen-Grossberg neural networks with continuously distributed
delays and impulses, Neurocomputing 73 (2010) 3105-3113.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:52231", author = "Meng Hu and Lili Wang", title = "Existence and Exponential Stability of Almost Periodic Solution for Cohen-Grossberg SICNNs with Impulses", abstract = "In this paper, based on the estimation of the Cauchy matrix of linear impulsive differential equations, by using Banach fixed point theorem and Gronwall-Bellman-s inequality, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays and impulses. An example is given to illustrate the main results.
", keywords = "Almost periodic solution, exponential stability, neural networks, impulses.", volume = "5", number = "4", pages = "534-10", }
