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Stability of impulsive delay differential equations with impulses at variable times. (English) Zbl 1088.34069

In the theory of impulsive differential equations, when the impulses take place at fixed moments, the results are easier to obtain by means of the corresponding results in the continuous case. But when the impulses are variable, there arise a number of difficulties related to the phenomena of “beating” of the solutions, bifurcation, loss of the property of autonomy, etc. However, from point of view both of the theory and applications, it is more natural to add impulses in the state variables rather than to add impulses at fixed times. The theory of impulsive differential equations with variable time impulses and delay develops rather slowly, in spite of some potential applications. There are only few results for uniqueness, stability and boundedness of the solutions. The present paper is a contribution to stability theory of such systems. By means of Lyapunov functions and the Razumikhin technique, some sufficient conditions are obtained for the uniform stability and uniform asymptotic stability of the zero solution of a delay differential equation with impulses at variable times.

MSC:

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

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