Stability of impulsive functional differential equations. (English) Zbl 1152.34053

This paper deals with the stability of impulsive functional differential equation in which the impulses depend on the delay. The authors obtain some stability results by means of Lyapunov functions and the Razumikhin technique. The work is illustrated by some examples.


34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
Full Text: DOI


[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Ballinger, G.; Liu, X. Z., Existence and uniqueness results for impulsive delay differential equations, Dyn. Contin. Discrete Impuls. Syst., 5, 579-591 (1999) · Zbl 0955.34068
[3] Rachunkova, I., Singular Dirichlet second-order BVPs with impulses, J. Differential Equations, 193, 435-459 (2003) · Zbl 1039.34015
[4] Shen, J. H., Razumikhin techniques in impulsive functional differential equations, Nonlinear Anal., 36, 1, 119-130 (1999) · Zbl 0939.34071
[5] Luo, Z. G.; Shen, J. H., New Razumikhin type theorems for impulsive functional differential equations, Appl. Math. Comput., 125, 375-386 (2002) · Zbl 1030.34078
[6] Zhang, Y.; Sun, J. T., Eventual practical stability of impulsive differential equations with time delay in terms of two measurements, J. Comput. Appl. Math., 176, 1, 223-229 (2005) · Zbl 1063.34070
[7] Sun, J. T.; Zhang, Y. P.; Wu, Q. D., Less conservative conditions for asymptotic stability of impulsive control systems, IEEE Trans. Automat. Control, 48, 5, 829-831 (2003) · Zbl 1364.93691
[8] Wang, P. G.; Liu, X., New comparison principle and stability criteria for impulsive hybrid systems on time scales, Nonlinear Anal., 7, 5, 1096-1103 (2006) · Zbl 1163.34365
[9] Kryszewski, W.; Plaskacz, S., Periodic solutions to impulsive differential inclusions with constraints, Nonlinear Anal., 65, 9, 1794-1804 (2006) · Zbl 1107.34004
[10] Arutyunov, A.; Karamzin, D.; Pereira, F., A nondegenerate maximum principle for the impulse control problem with state constraints, SIAM J. Control Optim., 43, 5, 1812-1843 (2005) · Zbl 1116.49013
[11] Wu, H. J.; Sun, J. T., \(p\)-moment stability of stochastic differential equations with impulsive jump and Markovian switching, Automatica, 42, 10, 1753-1759 (2006) · Zbl 1114.93092
[12] Ahmed, N. U., Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42, 2, 669-685 (2003) · Zbl 1037.49036
[13] Cobb, D., State feedback impulse elimination for singular systems over a Hermite domain, SIAM J. Control Optim., 44, 6, 2189-2209 (2006) · Zbl 1125.93013
[14] Bernfeld, S. R.; Corduneanu, C.; Ignatyev, A. O., On the stability of invariant sets of functional equations, Nonlinear Anal., 55, 641-656 (2003) · Zbl 1044.34027
[15] Martynyuk, A. A., Matrix-valued functionals approach for stability analysis of functional differential equations, Nonlinear Anal., 56, 793-802 (2004) · Zbl 1046.34087
[16] Lakshmikantham, V., Uniform asymptotic stability criteria for functional differential equations in terms of two measures, Nonlinear Anal., 34, 1, 1-6 (1998) · Zbl 0934.34064
[17] Han, M. A., Bifurcations of periodic solutions of delay differential equations, J. Differential Equations, 189, 396-411 (2003) · Zbl 1027.34081
[18] Hua, C. C.; Guan, X. P.; Shi, P., Robust backstepping control for a class of time delayed systems, IEEE Trans. Automat. Control, 50, 6, 894-899 (2005) · Zbl 1365.93054
[19] Morin, P.; Samson, C., Practical and asymptotic stabilization of chained systems by the transverse function control approach, SIAM J. Control Optim., 43, 1, 32-57 (2004) · Zbl 1101.93016
[20] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A., Practical Stability of Nonlinear Systems (1990), World Scientific: World Scientific Singapore · Zbl 0753.34037
[21] Martynyuk, A. A.; Shen, J. N.; Stavroulakis, I. P., Stability theorems in impulsive equations with infinite delay, (Advances in Stability Theory at the End of the 20th Century, vol. 13 (2003), Taylor and Francis: Taylor and Francis London, New York), 153-174 · Zbl 1047.34087
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