Existence and global exponential stability of equilibrium for impulsive neural network models with generalized piecewise constant delay. (English) Zbl 1485.93483
Summary: In this paper, we investigate the models of the impulsive cellular neural network with generalized constant piecewise delay (IDEGPCD). To guarantee the existence, uniqueness and global exponential stability of the equilibrium state, several new adequate conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEGPCD’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of the proposed results. We end the paper with a brief conclusion.
MSC:
| 93D23 | Exponential stability |
| 93C27 | Impulsive control/observation systems |
| 93B70 | Networked control |
| 93C43 | Delay control/observation systems |
Keywords:
impulsive neural networks; piecewise constant delay of generalized type; equilibrium; global exponential stability; Gronwall’s integral inequalityReferences:
| [1] | Akhmet, M. and Yımaz, E., Impulsive Hopfield-type neural network system with piecewise constant argument, Nonlinear Anal. Real World Appl.11 (2010) 2584-2593. · Zbl 1202.92001 |
| [2] | Akhmet, M. and Yımaz, E., Equilibria of Neural Networks with Impact Activations and Piecewise Constant Argument, in Neural Networks with Discontinuous/Impact Activations: Nonlinear Systems and Complexity, Vol. 9 (Springer, New York, 2014). · Zbl 1295.92007 |
| [3] | Barone, E. and Tebaldi, C., Stability of equilibria in a neural network model, Math. Methods Appl. Sci.23 (2000) 1179-1193. · Zbl 0959.92002 |
| [4] | Busenberg, S. and Cooke, K., Vertically Transmitted Diseases: Models and Dynamics in Biomathematics, Vol. 23 (Springer-Verlag, Berlin, 1993). · Zbl 0837.92021 |
| [5] | Cao, J., Global asymptotic stability of neural networks with transmission delays, Int. J. Syst. Sci.31 (2000) 1313-1316. · Zbl 1080.93517 |
| [6] | Chen, T., Global exponential stability of delayed Hopfield neural networks, Neural Netw.14 (2001) 977-980. |
| [7] | Chiu, K.-S. and Pinto, M., Variation of parameters formula and Gronwall’s inequality for differential equations with a general piecewise constant argument, Acta Math. Appl. Sin. Engl. Ser.27(4) (2011) 561-568. · Zbl 1270.34169 |
| [8] | Chiu, K.-S. and Pinto, M., Periodic solutions of differential equations with a general piecewise constant argument and applications, Electron. J. Qual. Theory Differ. Equ.46 (2010) 1-19. · Zbl 1211.34082 |
| [9] | Chiu, K.-S., Pinto, M. and Jeng, J.-Ch., Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument, Acta Appl. Math.133 (2014) 133-152. · Zbl 1315.34073 |
| [10] | Chiu, K.-S., Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstr. Appl. Anal.2013 (2013), Article ID: 196139, 13 pp., https://doi.org/10.1155/2013/196139. · Zbl 1298.34136 |
| [11] | Chiu, K.-S., On generalized impulsive piecewise constant delay differential equations, Sci. China Math.58 (2015) 1981-2002. · Zbl 1333.34118 |
| [12] | Chiu, K.-S. and Jeng, J.-Ch., Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr.288 (2015) 1085-1097. · Zbl 1329.34115 |
| [13] | Chiu, K.-S., Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math.151 (2017) 199-226. · Zbl 1373.92010 |
| [14] | Chiu, K.-S., Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci.38 (2018) 220-236. · Zbl 1399.34183 |
| [15] | Chiu, K.-S. and Li, T., Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr.292 (2019) 2153-2164. · Zbl 1442.34104 |
| [16] | Chiu, K.-S., Green’s function for periodic solutions in alternately advanced and delayed differential systems, Acta Math. Appl. Sin. Engl. Ser.36(4) (2020) 936-951. · Zbl 1472.34129 |
| [17] | Chiu, K.-S., Green’s function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications, Commun. Fac. Sci. Univ. Ank. Ser.A1 Math. Stat.70(1) (2021) 15-37. · Zbl 1489.34031 |
| [18] | Chua, L. O. and Yang, L., Cellular neural networks: Theory, IEEE Trans. Circuits Syst.35 (1988) 1257-1272. · Zbl 0663.94022 |
| [19] | Gopalsamy, K., Stability of artificial neural networks with impulses, Appl. Math. Comput.154 (2004) 783-813. · Zbl 1058.34008 |
| [20] | Huang, Z. K., Wang, X. H. and Gao, F., The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A350 (2006) 182-191. · Zbl 1195.34066 |
| [21] | Kwona, O. M., Lee, S. M., Park, Ju H. and Cha, E. J., New approaches on stability criteria for neural networks with interval time-varying delays, Appl. Math. Comput.218 (2012) 9953-9964. · Zbl 1253.34066 |
| [22] | Li, T., Yao, X., Wu, L. and Li, J., Improved delay-dependent stability results of recurrent neural networks, Appl. Math. Comput.218 (2012) 9983-9991. · Zbl 1253.34067 |
| [23] | Liu, Z., and Liao, L., Existence and global exponential stability of periodic solutions of cellular neural networks with time-varying delays, J. Math. Anal. Appl.290 (2004) 247-262. · Zbl 1055.34135 |
| [24] | Lou, X. Y. and Cui, B. T., Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl.330 (2007) 144-158. · Zbl 1111.68104 |
| [25] | Mohamad, S. and Gopalsamy, K., Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput.135 (2003) 17-38. · Zbl 1030.34072 |
| [26] | Park, J. H., Global exponential stability of cellular neural networks with variable delays, Appl. Math. Comput.183 (2006) 1214-1219. · Zbl 1115.34071 |
| [27] | Pinto, M., Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments, Math. Comp. Model.49 (2009) 1750-1758. · Zbl 1171.34321 |
| [28] | Pinto, M., Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl.17(2) (2011) 235-254. · Zbl 1220.34082 |
| [29] | Shah, S. M. and Wiener, J., Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci.6 (1983) 671-703. · Zbl 0534.34067 |
| [30] | Wang, B., Zhong, S. and Liu, X., Asymptotical stability criterion on neural networks with multiple time-varying delays, Appl. Math. Comput.195 (2008) 809-818. · Zbl 1137.34357 |
| [31] | Wiener, J., Differential equations with piecewise constant delays, Trends in the Theory and Practice of Nonlinear Differential Equations, eds. Lakshmikantham, V. and Dekker, M. (CRC press, New York, 1983), pp. 547-580. |
| [32] | Wiener, J., Generalized Solutions of Functional Differential Equations (World Scientific, Singapore, 1993). · Zbl 0874.34054 |
| [33] | Wiener, J. and Lakshmikantham, V., Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud.7 (2000) 60-69. · Zbl 0997.34076 |
| [34] | Xu, B., Liu, X. and Liao, X., Global exponential stability of high order Hopfield type neural networks, Appl. Math. Comput.174 (2006) 98-116. · Zbl 1099.34051 |
| [35] | Zhou, L. and Hu, G., Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput.195 (2008) 402-411. · Zbl 1137.34350 |
| [36] | Zhang, Y., Yue, D. and Tian, E., New stability criteria of neural networks with interval time-varying delay: A piecewise delay method, Appl. Math. Comput.208 (2009) 249-259. · Zbl 1171.34048 |
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