×

Practical stability of impulsive discrete systems with time delays. (English) Zbl 1474.93150

Summary: The purpose of this paper is to investigate the practical stability problem for impulsive discrete systems with time delays. By using Lyapunov functions and the Razumikhin-type technique, some criteria which guarantee the practical stability and uniformly asymptotically practical stability of the addressed systems are provided. Finally, two examples are presented to illustrate the criteria.

MSC:

93C55 Discrete-time control/observation systems
93D20 Asymptotic stability in control theory
39A30 Stability theory for difference equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Zhang, Z.; Liu, X., Robust stability of uncertain discrete impulsive switching systems, Computers & Mathematics with Applications, 58, 2, 380-389, (2009) · Zbl 1189.39027
[2] Song, Q.; Cao, J., Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses, Journal of the Franklin Institute, 345, 1, 39-59, (2008) · Zbl 1167.93369
[3] Nigmatulin, R.; Kipnis, M., Stability of the discrete population model with two delays, Proceedings of the International Conference on Physics and Control, IEEE
[4] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0978.39001
[5] Zhang, Y.; Sun, J.; Feng, G., Impulsive control of discrete systems with time delay, IEEE Transactions on Automatic Control, 54, 4, 871-875, (2009) · Zbl 1367.93344
[6] Liu, B.; Marquez, H. J., Razumikhin-type stability theorems for discrete delay systems, Automatica, 43, 7, 1219-1225, (2007) · Zbl 1123.93065
[7] Liu, X.; Zhang, Z., Uniform asymptotic stability of impulsive discrete systems with time delay, Nonlinear Analysis: Theory, Methods & Applications, 74, 15, 4941-4950, (2011) · Zbl 1225.39022
[8] Zhang, Y., Exponential stability of impulsive discrete systems with time delays, Applied Mathematics Letters, 25, 12, 2290-2297, (2012) · Zbl 1252.39024
[9] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A., Practical Stability of Nonlinear Systems, (1990), Singapore: World Scientific, Singapore · Zbl 0753.34037
[10] Kou, C. H.; Zhang, S. N., Practical stability for finite delay differential systems in terms of two measures, Acta Mathematicae Applicatae Sinica, 25, 3, 476-483, (2002) · Zbl 1049.34092
[11] Bainov, D. D.; Stamova, I. M., On the practical stability of the solutions of impulsive systems of differential-difference equations with variable impulsive perturbations, Journal of Mathematical Analysis and Applications, 200, 2, 272-288, (1996) · Zbl 0848.34058
[12] Zhang, Y.; Sun, J., Practical stability of impulsive functional differential equations in terms of two measurements, Computers & Mathematics with Applications, 48, 10-11, 1549-1556, (2004) · Zbl 1075.34083
[13] Zhang, Y.; Sun, J., Eventual practical stability of impulsive differential equations with time delay in terms of two measurements, Journal of Computational and Applied Mathematics, 176, 1, 223-229, (2005) · Zbl 1063.34070
[14] Villafuerte, R.; Mondié, S.; Poznyak, A., Practical stability of time-delay systems: LMI’s approach, European Journal of Control, 17, 2, 127-138, (2011) · Zbl 1229.93092
[15] Stamova, I. M., Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 325, 1, 612-623, (2007) · Zbl 1113.34058
[16] Baĭnov, D. D.; Simeonov, P. S., Systems with Impulse Effect: Stability, Theory and Applications, (1989), New York, NY, USA: Halsted Press, New York, NY, USA · Zbl 0676.34035
[17] Lakshmikantham, V.; Baĭnov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations. Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, (1989), Singapore: World Scientific, Singapore · Zbl 0719.34002
[18] Yang, T., Impulsive Systems and Control: Theory and Applications, (2001), Huntington, NY, USA: Nova Science Publishers, Huntington, NY, USA
[19] Fu, X.; Yan, B.; Liu, Y., Introduction of Impulsive Differential Systems, (2005), Beijing, China: Science Press, Beijing, China
[20] Lakshmikantham, V.; Matrosov, V. M.; Sivasundaram, S., Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Mathematics and Its Applications, 63, (1991), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0721.34054
[21] Shen, J. H., Razumikhin techniques in impulsive functional-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 36, 1, 119-130, (1999) · Zbl 0939.34071
[22] Luo, Z.; Shen, J., New Razumikhin type theorems for impulsive functional differential equations, Applied Mathematics and Computation, 125, 2-3, 375-386, (2002) · Zbl 1030.34078
[23] Li, X., Further analysis on uniform stability of impulsive infinite delay differential equations, Applied Mathematics Letters, 25, 2, 133-137, (2012) · Zbl 1236.34098
[24] Fu, X.; Li, X., Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems, Journal of Computational and Applied Mathematics, 224, 1, 1-10, (2009) · Zbl 1179.34079
[25] Li, X., New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays, Nonlinear Analysis: Real World Applications, 11, 5, 4194-4201, (2010) · Zbl 1210.34103
[26] Luo, Z.; Shen, J., Stability of impulsive functional differential equations via the Liapunov functional, Applied Mathematics Letters, 22, 2, 163-169, (2009)
[27] Liu, X.; Wang, Q., The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear Analysis: Theory, Methods & Applications, 66, 7, 1465-1484, (2007) · Zbl 1123.34065
[28] Li, X.; Bohner, M., An impulsive delay differential inequality and applications, Computers & Mathematics with Applications, 64, 6, 1875-1881, (2012) · Zbl 1268.34159
[29] Li, X., Uniform asymptotic stability and global stabiliy of impulsive infinite delay differential equations, Nonlinear Analysis: Theory, Methods & Applications, 70, 5, 1975-1983, (2009) · Zbl 1175.34094
[30] Li, X.; Akca, H.; Fu, X., Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions, Applied Mathematics and Computation, 219, 14, 7329-7337, (2013) · Zbl 1297.34082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.