Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions. (English) Zbl 1216.34011

The authors develop the concepts of stability, practical stability and boundedness in terms of two measures. The object is a system of impulsive differential equations to which the method of perturbing Lyapunov functions is applied.


34A37 Ordinary differential equations with impulses
34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
Full Text: DOI


[1] Liu, X. Z.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal., 53, 1041-1062 (2003) · Zbl 1037.34061
[2] Gao, C. X.; Li, K. Z.; Feng, E. M., Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos Solitons Fractals, 28, 271-277 (2006) · Zbl 1079.92036
[3] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[4] Akinyele, O., Cone-valued Lyapunov functions and stability of impulsive control systems, Nonlinear Anal., 39, 247-259 (2000) · Zbl 0939.34057
[5] Ahmed, N. U., Some remarks on the dynamics of impulsive systems in Banach spaces, Dyn. Contin. Discrete Impuls. Syst., 8, 261-274 (2001) · Zbl 0995.34050
[6] Ahmed, N. U.; Teo, K. L.; Hou, S. H., Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Anal., 54, 907-925 (2003) · Zbl 1030.34056
[7] Akhmet, M. U., On the general problem of stability fro impulsive differential equations, J. Math. Anal. Appl., 288, 182-196 (2003) · Zbl 1047.34094
[8] Ma, Y. F.; Xiu, Z. L.; Sun, L. H.; Feng, E. M., Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay, Int. J. Nonlinear Sci. Numer. Simul., 7, 3, 305-308 (2006)
[9] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A., Practical Stability of Nonlinear Systems (1990), World Scientific: World Scientific Singapore · Zbl 0753.34037
[10] McRae, F. A., Perturbing Lyapunov functions and stability criteria for initial time difference, Appl. Math. Comput., 117, 313-320 (2001) · Zbl 1089.70013
[11] Zhang, Y.; Sun, J. T., Practical stability of impulsive functional differential equations in terms of two measures, Comput. Math. Appl., 48, 1549-1556 (2004) · Zbl 1075.34083
[12] Zhai, G. S.; Michel, A. N., Generalized practical stability analysis of discontinuous dynamical system, Int. J. Appl. Math. Comput. Sci., 14, 1, 5-12 (2004) · Zbl 1171.34328
[13] Stamova, I. M., Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations, J. Math. Anal. Appl., 325, 612-623 (2007) · Zbl 1113.34058
[14] Stutson, D.; Vatsala, A. S., Composite boundedness and stability results by perturbing Lyapunov functions, Nonlinear Anal., 26, 761-766 (1994) · Zbl 0842.34057
[15] Soliman, A. A., On perturbing Liapunov functional, Appl. Math. Comput., 133, 319-325 (2004) · Zbl 1030.34071
[16] Soliman, A. A., On cone perturbing Liapunov function for impulsive differential systems, Appl. Math. Comput., 163, 1069-1079 (2005) · Zbl 1068.34049
[17] Zhao, H. Q.; Feng, E. M., \( \phi_0\)-Stability of an impulsive system obtained from perturbing Lyapunov functions, Nonlinear Anal., 66, 962-967 (2007) · Zbl 1115.34052
[18] Zhao, H. Q.; Feng, E. M., Stability of impulsive system by perturbing Lyapunov functions, Appl. Math. Lett., 20, 194-198 (2007) · Zbl 1113.34305
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