The method of Lyapunov functionals and exponential stability of impulsive systems with time delay. (English) Zbl 1123.34065

Consider the system of functional differential equations with impulses \[ x'(t)=f(t,x_t),\, t\neq t_k, \]
\[ x(t_k^+)-x(t_k^-)=I_k(t_k,x_{t_k^-}),\, k\in\mathbb{N}, \]
\[ x_{t_0}=\phi, \] where, as usual, \(x_t(s)=x(t+s),\, s\in[-\tau,0]\). It is assumed that zero is a solution to this system.
Using Lyapunov-like functionals, sufficient conditions are found to prove that the trivial solution is exponentially stable. The paper also contains several examples; in particular, it is shown that an unstable linear delay-differential equation may become exponentially stable by adding a suitable impulsive (nondelayed) perturbation.
Reviewer: Eduardo Liz (Vigo)


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


[1] Anokhin, A.; Berezansky, L.; Braverman, E., exponential stability of linear delay impulsive differential equations, J. Math. Anal. Appl., 193, 923-941 (1995) · Zbl 0837.34076
[2] Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dyn. Contin. Discrete Impuls. Syst., 5, 579-591 (1999) · Zbl 0955.34068
[3] Ballinger, G.; Liu, X., Practical stability of impulsive delay differential equations and applications to control problems, (Optimization Methods and Applications (2001), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht) · Zbl 0879.34015
[4] Berezansky, L.; Idels, L., Exponential stability of some scalar impulsive delay differential equation, Commun. Appl. Math. Anal., 2, 301-309 (1998) · Zbl 0901.34068
[5] Gopalsamy, K.; Zhang, B. G., On delay differential equations with impulses, J. Math. Anal. Appl., 139, 110-122 (1989) · Zbl 0687.34065
[6] Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[7] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional differential equations (1986), Academic Press Inc.: Academic Press Inc. London · Zbl 0593.34070
[8] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[9] Lakshmikantham, V.; Liu, X., Stability criteria for impulsive differential equations in terms of two measures, J. Math. Anal. Appl., 137, 591-604 (1989) · Zbl 0688.34031
[10] Li, X.; Weng, P., Impulsive stabilization of two kinds of second-order linear delay differential equations, J. Math. Anal. Appl., 291, 270-281 (2004) · Zbl 1047.34098
[11] Liu, X., Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10, 11-19 (1993) · Zbl 0789.93101
[12] Liu, X., Stability results for impulsive differential systems with applications to population growth models, Dyn. Stability Syst., 9, 163-174 (1994) · Zbl 0808.34056
[13] Liu, X.; Ballinger, G., Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal., 51, 633-647 (2002) · Zbl 1015.34069
[14] Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal., 53, 1041-1062 (2003) · Zbl 1037.34061
[15] Liu, X.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41, 903-915 (2001) · Zbl 0989.34061
[16] Liu, X.; Shen, X.; Zhang, Y., exponential stability of singularly perturbed systems with time delay, Appl. Anal., 82, 117-130 (2003) · Zbl 1044.34031
[17] Wang, Q.; Liu, X., Exponential stability for impulsive delay differential equations by Razumikhin method, J. Math. Anal. Appl., 309, 462-473 (2005) · Zbl 1084.34066
[18] Shen, J.; Luo, Z.; Liu, X., Impulsive stabilization of functional differential equations via Liapunov functionals, J. Math. Anal. Appl., 240, 1-5 (1999) · Zbl 0955.34069
[19] Stamova, I. M.; Stamov, G. T., Lyapunov-Razumikhin method for impulsive functional equations and applications to the population dynamics, J. Comput. Appl. Math., 130, 163-171 (2001) · Zbl 1022.34070
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