On Retention of Eventual Stability of Perturbed Impulsive Differential Systems

Anju Sood, Sanjay Kumar Srivastava


In this paper, a system of non nonlinear differential equations with
impulse effect at fixed time moments is considered and criteria for retention of
uniform eventual stability of its perturbed impulsive differential systems under
vanishing perturbations are established. Sufficient conditions are obtained by
using piecewise continuous Lyapunov functions. An example is also worked out
to illustrate the results.

AMS Subject Classification: 34CXX, 34DXX, 34A37, 34K45.


Eventual stability; impulsive differential systems; Lyapunov function; uniform eventual stability; perturbed differential systems

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Kulev, G.K. & Bainov, D.D., Stability of Systems with Impulses by the Direct Method of Lyapunov, Bull Austral. Math. Soc., 38, pp. 113-123, 1988.

Lakshmikantham, V., Bainov, D.D. & Simeonov, P.S., Theory of Impulsive Differential Equations, World Scientific: Singapore, 6, pp. 102-194, 1989.

Lakshmikantham, V., Leela, S. & Martynyuk, A.A., Practical Stability of Nonlinear Systems, World Scientific: Singapore, 1990.

Soliman, A.A., Stability Criteria of Impulsive Differential Systems, Applied Mathematics and Computation, 134, pp. 445-457, 2003.

Zhang, Y. & Jitao, S., Eventual Stability of Impulsive Differential Systems, Acta Mathematica Scientia, 27B(2), pp. 373-380, 2007.

Sood, A. & Srivastava, S.K., Ψ-Eventual Stability of Differential Systems with Impulses, Global Journal of Science Frontier Research: Mathematics & Decision Sciences, 14(6), pp. 1-8, 2014.

Gladilina, R.I. & Ignat’ev, A.O., On Retention of Impulsive System Stability under Perturbations, Automation and Remote Control, 68, pp. 1364-1371, 2007.

Vassalos, D., Hamamoto, M., Molyneux, D. & Papanikolaou, A., Contemporary Ideas on Ship Stability, Elsevier Science Ltd: U.K., 1st Edition, 2000.

Lakshmikantham, V. & Leela, S., Differential and Integral Inequalities- Theory and Applications, Academic Press: New York, 1, pp. 131-190, 1969.

Xinzhi, Liu & Sivasundram, S., Stability of Nonlinear Systems Under Constantly Acing Perturbations, Internat. J. Math. & Math. Sci., 18(2), pp. 273-278, 1995.

Pandit, S.G., Differential Systems with Impulsive Perturbations, Pacific Journal of Mathematics, 86(2), pp. 553-560, 1989.

Cantarelli, G. & Zappala, G., Stability Properties of Differential Systems Under Constantly Acting Perturbations, Electronic Journal of Differential Equations, 152, pp. 1-16, 2010.

Sheldon, P. & Gordon, A., Stability Theory for Perturbed Differential Equations, 2(2), pp. 283-297, 1996.

Andreev, A. & Zappala, G., On Stability for Perturbed Differential Equations, La Matematiche, L1-Fasc. I, pp. 27-41, 1996.

Soliman, A.A., On Stability of Perturbed Impulsive Differential Systems, Applied Mathematics and Computation, 133, pp.105-117, 2002.

Kulev, G.K., Uniform asymptotic Stability in Impulsive Perturbed Systems of Differential Equations, Journal of Computational and Applied Mathematics, 41, pp. 49-55, 1992.

DOI: http://dx.doi.org/10.5614%2Fj.math.fund.sci.2016.48.1.1


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