## Exponential stability for impulsive delay differential equations by Razumikhin method.(English)Zbl 1084.34066

This paper deals with the exponential stablity of the solutions of the following impulsive delay differential equation $\dot x(t)=f(t,x_t), \;\;t\not=t_{k},$
$\Delta x(t)=I_{k}(t,x_{t^{-}}), \;t=t_{k}, \;k\in \mathbb{N},$
$x_{t_{0}}=\phi.$ The proofs are based on the Razumikhin method. Some examples illustrating the results are presented.

### MSC:

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

### References:

 [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, Dynam. contin. discrete impuls. systems, 5, 579-591, (1999) · Zbl 0955.34068 [3] Ballinger, G.; Liu, X., Practical stability of impulsive delay differential equations and applications to control problems, () · Zbl 0879.34015 [4] Berezansky, L.; Idels, L., Exponential stability of some scalar impulsive delay differential equation, Comm. 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 New York [7] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [8] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press London · Zbl 0593.34070 [9] Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Stability analysis of nonlinear systems, (1989), Dekker New York · Zbl 0676.34003 [10] 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 [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, Dynam. stability systems, 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] Liu, Y.; Ge, W., Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear anal., 57, 363-399, (2004) · Zbl 1064.34051 [18] Shen, J.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear anal., 33, 519-531, (1998) · Zbl 0933.34083 [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 [20] Yan, J.; Shen, J., Impulsive stabilization of functional differential equations by lyapunov – razumikhin functions, Nonlinear anal., 37, 245-255, (1999) · Zbl 0951.34049
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.