×

Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems. (English) Zbl 1179.34079

The authors consider the impulsive FDE of the form \[ \begin{split} x'(t)&=f(t,x(\cdot)),\;\;t\neq t_k,\\ \left.\Delta\right|_{t=t_k}&=x(t_k)-x(t_k^-)=I_k(t_k,x(t_k^-)), \;\;k=1,2,\ldots, \end{split} \] \(0\leq t_0<t_1<\cdots\), \(x'\) denotes the right-hand derivative, \(f\) is a continuous functional defined in the appropriate space, \(f(t,0)=I_k(t_k,0)=0\). It is supposed that the IVP has a unique solution \(x(t,\sigma,\phi)\) which can be continued to \(\infty\). The initial function \(\phi\) is piecewise continuous.
The zero solution is said to be weak exponentially stable if for any \(\varepsilon>0\) and \(\sigma\geq t_0\) \(\exists\delta>0\) such that \(\|\phi\|<\delta\) implies \(\alpha(\|x(t,\sigma,\phi)\|)<\varepsilon e^{-\lambda(t-\sigma)}\) for \(t\geq\sigma\) and some \(\lambda>0\) and a strictly increasing \(\alpha: \mathbb{R}_+\to \mathbb{R}_+\). If \(\alpha(s)=s\) we obtain the exponential stability.
Two theorems on the weak exponential stability are proved. The paper ends with two illustrative examples. There are many misprints.

MSC:

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

References:

[1] Bainov, D.D.; Simennov, P.S., Systems with impulsive effect stability theory and applications, (1989), Ellis Horwood Limited New York
[2] Fu, X.; Yan, B.; Liu, Y., Introduction of impulsive differential systems, (2005), Science Press Beijing
[3] Xing, Y.; Han, M., A new approach to stability of impulsive functional differential equations, Applied mathematics and computation, 151, 835-847, (2004) · Zbl 1057.34103
[4] Yang, T., Impulsive systems and control: theory and applications, (2001), Nova Science Publishers, Inc Huntington, NY
[5] Luo, Z.; Shen, J., Stability results for impulsive functional differential equations with infinite delays, Journal of computational and applied mathematics, 131, 55-64, (2001) · Zbl 0988.34059
[6] Fu, X.; Liu, X., Uniform boundedness and stability criteria in terms of two measures for impulsive integro- differential equations, Applied mathematics and computation, 102, 237-255, (1999) · Zbl 0929.45005
[7] Soliman, A.A., Stability criteria of impulsive differential systems, Applied mathematics and computation, 134, 445-457, (2003) · Zbl 1030.34046
[8] Luo, Z.; Shen, J., Impulsive stabilization of functional differential equations with infinite delays, Applied mathematics letters, 16, 695-701, (2003) · Zbl 1068.93054
[9] Liu, B.; Liu, X.; Teo, K.; Wang, Q., Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA journal of applied mathematics, 71, 47-61, (2006) · Zbl 1128.34047
[10] Wang, Q.; Liu, X., Exponential stability for impulsive delay differential equations by Razumikhin method, Journal of mathematical analysis and application, 309, 462-473, (2005) · Zbl 1084.34066
[11] Anokhin, A.; Berezansky, L.; Braverman, E., Exponential stability of linear delay impulsive differential equations, Journal of mathematical analysis and application, 193, 923-941, (1995) · Zbl 0837.34076
[12] Shen, J.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear analysis, 33, 519-531, (1998) · Zbl 0933.34083
[13] Ballinger, G.; Lui, X., Existence, uniqueness and boundedness results for impulsive delay differential equations, Applicable analysis, 74, 71-93, (2000) · Zbl 1031.34081
[14] Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear analysis, 53, 1041-1062, (2003) · Zbl 1037.34061
[15] Shen, J., Existence and uniqueness of solutions for a class of infinite delay functional differential equations with applications to impulsive differential equations, Journal of huaihua teacher’s college, 15, 45-51, (1996)
[16] Berezansky, L.; Idels, L., Exponential stability of some scalar impulsive delay differential equation, Communications in applied mathematics analysis, 2, 301-309, (1998) · Zbl 0901.34068
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.