Mean almost periodicity and moment exponential stability of discrete-time stochastic shunting inhibitory cellular neural networks with time delays

Zhang, Tianwei ; Xu, Lijun

Kybernetika, Tome 55 (2019) no. 4, pp. 690-713

Abstract (VO)

By using the semi-discrete method of differential equations, a new version of discrete analogue of stochastic shunting inhibitory cellular neural networks (SICNNs) is formulated, which gives a more accurate characterization for continuous-time stochastic SICNNs than that by Euler scheme. Firstly, the existence of the 2th mean almost periodic sequence solution of the discrete-time stochastic SICNNs is investigated with the help of Minkowski inequality, Hölder inequality and Krasnoselskii's fixed point theorem. Secondly, the moment global exponential stability of the discrete-time stochastic SICNNs is also studied by using some analytical skills and the proof of contradiction. Finally, two examples are given to demonstrate that our results are feasible. By numerical simulations, we discuss the effect of stochastic perturbation on the almost periodicity and global exponential stability of the discrete-time stochastic SICNNs.

Détail

DOI : 10.14736/kyb-2019-4-0690

Classification : 39A24, 39A30, 39A50, 92B20
Keywords: semi-discrete method; stochastic; Krasnoselskii's fixed point theorem; almost periodicity; global exponential stability

Bibliographie

[1] Arnold, L., Tudor, C.: Stationary and almost periodic solutions of almost periodic affine stochastic differential equations. Stochastics and Stochast. Reports 64 (1998), 177-193. | DOI | MR

[2] Arunkumar, A., Sakthivel, R., Mathiyalagan, K., Park, J. H.: Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks. ISA Transactions 53 (2014), 1006-1014. | DOI

[3] Bashkirtseva, I.: Controlling the stochastic sensitivity in thermochemical systems under incomplete information. Kybernetika 54 (2018), 96-109. | DOI | MR

[4] Bezandry, P. H., Diagana, T.: Almost Periodic Stochastic Processes. Springer, New York 2011. | DOI | MR

[5] Bouzerdoum, A., Pinter, R. B.: Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Trans. Circuits Systems 1 - Fundament. Theory Appl. 40 (1993), 215-221. | DOI | MR

[6] Chen, A., Cao, J., Huang, L.: Almost periodic solution of shunting inhibitory CNNs with delays. Phys. Lett. A 298 (2002), 161-170. | DOI | MR

[7] Dorogovtsev, A. Y., Ortega, O. A.: On the existence of periodic solutions of a stochastic equation in a Hilbert space. Visnik Kiiv. Univ. Ser. Mat. Mekh. 30 (1988), 21-30. | MR

[8] Du, Bo, Liu, Y. R., Abbas, I. A.: Existence and asymptotic behavior results of periodic solution for discrete-time neutral-type neural networks. J. Frank. Inst. 353 (2016), 448-461. | DOI | MR

[9] Fan, Q. Y., Shao, J. Y.: Positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying and continuously distributed delays. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 1655-1663. | DOI | MR

[10] Hu, M. F., Cao, J. D., Hua, A. H.: Mean square exponential stability for discrete-time stochastic switched static neural networks with randomly occurring nonlinearities and stochastic delay. Neurocomputing 129 (2014), 476-481. | DOI | MR

[11] Hu, S. G., Huang, C. M., Wu, F. K.: Stochastic Differential Equations. Science Press, Beijing 2008. | DOI | MR

[12] Hu, S., Wang, J.: Global robust stability of a class of discrete-time interval neural networks. IEEE Trans. Circuits Syst. I Regul. Pap. 53 (2006), 129-138. | DOI | MR

[13] Huang, Z. K., Mohamad, S., Gao, F.: Multi-almost periodicity in semi-discretizations of a general class of neural networks. Math. Computers Simul. 101 (2014), 43-60. | DOI | MR

[14] Huang, Z. K., Wang, X. H., Gao, F.: The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks. Physics Lett. A 350 (2006), 182-191. | DOI | MR

[15] Huang, Z. K., Wang, X. H., Xia, Y. H.: Exponential attractor of $\kappa$-almost periodic sequence solution of discrete-time bidirectional neural networks. Simul. Modell. Practice Theory 18 (2010), 317-337. | DOI

[16] Kawata, T.: Almost Periodic Weakly Stationary Processes, Statistics and Probability: Essays in Honor of C. R. Rao. North-Holland, Amsterdam 1982, pp. 383-396. | MR

[17] Kuang, J. C.: Applied Inequalities. Shandong Science and Technology Press, Shandong 2012.

[18] Lan, Q. X., Niu, H. W., Liu, Y. M., Xu, H. F.: Global output-feedback finite-time stabilization for a class of stochastic nonlinear cascaded systems. Kybernetika 53 (2017), 780-802. | DOI | MR

[19] Liu, B., Huang, L.: Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays. Phys. Lett. A 349 (2006), 177-186. | DOI | MR

[20] Liu, D., Wang, L.J., Pan, Y. N., Ma, H. Y.: Mean square exponential stability for discrete-time stochastic fuzzy neural networks with mixed time-varying delay. Neurocomputing 171 (2016), 1622-1628. | DOI

[21] Mohamad, S.: Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks. Physica D 159 (2001), 233-251. | DOI | MR

[22] Mohamad, S., Gopalsamy, K.: Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Math. Computers Simul. 53 (2000), 1-39. | DOI | MR

[23] Mohamad, S., Gopalsamy, K.: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl. Math. Comput. 135 (2003), 17-38. | DOI | MR

[24] Mohamad, S., Naim, A. G.: Discrete-time analogues of integro-differential equations modelling bidirectional neural networks. J. Comput. Appl. Math. 138 (2002), 1-20. | DOI | MR

[25] Nagamani, G., Ramasamy, S., Balasubramaniam, P.: Robust dissipativity and passivity analysis for discrete-time stochastic neural networks with time-varying delay. Complexity 21 (2014), 47-58. | DOI | MR

[26] Ou, Y., Liu, H., Si, Y., Feng, Z.: Stability analysis of discrete-time stochastic neural networks with time-varying delays. Neurocomputing 73 (2010), 740-748. | DOI

[27] Raj, S., Ramachandran, R., Rajendiran, S., Cao, J. D.: Passivity analysis of uncertain stochastic neural network with leakage and distributed delays under impulsive perturbations. Kybernetika 54 (2018), 3-29. | DOI | MR

[28] Şaylı, M., Yılmaz, E.: Periodic solution for state-dependent impulsive shunting inhibitory CNNs with time-varying delays. Neural Networks 68 (2015), 1-11. | DOI

[29] Smart, D. R.: Fixed Point Theorems. Cambridge University Press, Cambridge 1980. | MR

[30] Swift, R. J.: Almost periodic harmonizable processes. Georgian Math. J. 3 (1996), 275-292. | DOI | MR

[31] Tudor, C.: Almost periodic solutions of affine stochastic evolutions equations. Stochastics and Stochast. Reports 38 (1992), 251-266. | DOI | MR

[32] Wang, P., Li, B., Li, Y. K.: Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing 167 (2015), 76-82. | DOI | MR

[33] Wang, J., Zhang, X. M., Han, Q. L.: Event-Triggered generalized dissipativity filtering for neural networks with time-varying delays. IEEE Trans. Neural Networks Learning Systems 27 (2016), 77-88. | DOI | MR

[34] Xiong, W., Cao, J. D.: Global exponential stability of discrete-time Cohen-Grossberg neural networks. Neurocomputing 64 (2005), 433-446. | DOI

[35] Xiong, L. L., Cheng, J., Cao, J. D., Liu, Z. X.: Novel inequality with application to improve the stability criterion for dynamical systems with two additive time-varying delays. Applied Math. Comput. 321 (2018), 672-688. | DOI | MR

[36] Xu, C. J., Li, P. L.: On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator. Neurocomputing 275 (2018), 377-382. | DOI

[37] Yang, L., Li, Y. K.: Periodic solutions for stochastic shunting inhibitory cellular neural networks with distributed delays. Advances Difference Equations 2014 (2014), 1-37. | DOI | MR

[38] Yao, L. G.: Global exponential convergence of neutral type shunting inhibitory cellular neural networks with d operator. Neural Process. Lett. 45 (2017), 401-409. | DOI

[39] Zhang, T. W.: Multiplicity of positive almost periodic solutions in a delayed Hassell-Varleytype predator-prey model with harvesting on prey. Math. Meth. Appl. Sci. 37 (2014), 686-697. | DOI | MR

[40] Zhang, T. W.: Almost periodic oscillations in a generalized Mackey-Glass model of respiratory dynamics with several delays. Int. J. Biomath. 7 (2014), 1450029 (22 pages). | DOI | MR

[41] Zhang, T. W., Gan, X. R.: Almost periodic solutions for a discrete fishing model with feedback control and time delays. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 150-163. | DOI | MR

[42] Zhang, X. M., Han, Q. L.: Global asymptotic stability for a class of generalized neural networks with interval time-varying delays. IEEE Trans. Neural Networks 22 (2011), 1180-1192. | DOI

[43] Zhang, X. M., Han, Q. L.: An overview of recent developments in Lyapunov-Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays. Neurocomputing 313 (2018), 392-401. | DOI

[44] Zhang, X. M., Han, Q. L., Zeng, Z. G.: Hierarchical type stability criteria for delayed neural networks via canonical Bessel-Legendre inequalities. IEEE Trans. Cybernet. 48 (2018), 1660-1671. | DOI

[45] Zhang, T. W., Liao, Y. Z.: Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay. Kybernetika 53 (2017), 612-629. | DOI | MR

[46] Zhang, H. Y., Qiu, Z. P., Xiong, L. L.: Stochastic stability criterion of neutral-type neural networks with additive time-varying delay and uncertain semi-Markov jump. Neurocomputing 333 (2019), 395-406. | DOI

[47] Zhang, T. W., Xiong, L. L.: Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Applied Mathematics Letters 101 (2020), 106072. | DOI | MR

[48] Zhang, T. W., Yang, L., Xu, L. J.: Stage-structured control on a class of predator-prey system in almost periodic environment. International Journal of Control 2019 (2019), in print. | DOI

[49] Zhang, Z. Q., Zhou, D. M.: Existence and global exponential stability of a periodic solution for a discrete-time interval general BAM neural networks. J. Frank. Inst. 347 (2010), 763-780. | DOI | MR

[50] Zhao, H., Sun, L., Wang, G.: Periodic oscillation of discrete-time bidirectional associative memory neural networks. Neurocomputing 70 (2007), 2924-2930. | DOI