×

zbMATH — the first resource for mathematics

Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method. (English) Zbl 1159.34347
The authors study the stability of nonlinear impulsive delay differential systems in the form \[ x'(t)=F(t,x_t),\;t\not=t_k;\quad \Delta x(t_k)=I_k(t_k, x_{t_k^-}),\;k\in\mathbb{N}; \quad x_{t_0}=\phi. \] By employing the Razumikhin technique and the Lyapunov function method, they obtain sufficient conditions for the trivial solution to be globally exponentially stable. Criteria for the global exponential stability of a linear impulsive delay system are derived as applications. An example and its simulation are given to illustrate the results. This nice paper will be of interest to the researchers working on impulsive differential equations and on the stability theory.

MSC:
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
PDF BibTeX XML Cite
Full Text: DOI
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, Dyn. contin. discrete impuls. syst., 5, 579-591, (1999) · Zbl 0955.34068
[3] Ballinger, G.; Liu, X.Z., Practical stability of impulsive delay differential equations and applications to control problems, ()
[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] Chen, T.P., Global exponential stability of delayed Hopfield neural networks, Neural networks, 14, 977-980, (2001)
[6] Liu, B.; Liu, X.Z.; Teo, K.; Wang, Q., Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. appl. math., 71, 47-61, (2006) · Zbl 1128.34047
[7] Liu, X.Z., Stability results for impulsive differential systems with applications to population growth models, Dyn. stab. syst., 9, 163-174, (1994) · Zbl 0808.34056
[8] Liu, X.Z., Impulsive stabilization of nonlinear systems, IMA J. math. control inform., 10, 11-19, (1993) · Zbl 0789.93101
[9] Liu, X.Z.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput. math. appl., 41, 903-915, (2001) · Zbl 0989.34061
[10] Shen, J.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear anal., 33, 519-531, (1998) · Zbl 0933.34083
[11] 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
[12] Wang, Q.; Liu, X.Z., Exponential stability for impulsive delay differential equations by Razumikhin method, J. math. anal. appl., 309, 462-473, (2005) · Zbl 1084.34066
[13] Luo, Z.; Shen, J., Impulsive stabilization of functional differential equations with infinite delays, Appl. math. lett., 16, 695-701, (2003) · Zbl 1068.93054
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.