Uniform practical stability of perturbed impulsive differential system in terms of two measures

Pallvi Mahajan; Sanjay Kumar Srivastava; Rakesh Dogra

doi:10.26637/MJM0702/0003

Abstract

In the present paper, a perturbed impulsive differential system is investigated for uniform practical stability. For the same, sufficient conditions have been obtained in terms of two measures by using Lyapunov-like function. An example has also been given to illustrate the derived results.

Keywords:

Perturbed Impulsive differential system, Practical Stability, Two Measures, Lyapunov function

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Pallvi Mahajan Inder Kumar Gujral-Punjab Technical University, Kapurthala-144601, Punjab, India.
Sanjay Kumar Srivastava Beant College of Engineering and Technology, Gurdaspur-143521, Punjab, India.
Rakesh Dogra Beant College of Engineering and Technology, Gurdaspur-143521, Punjab, India.

Pages: 142-146
Date Published: 01-04-2019
Vol. 7 No. 02 (2019): Malaya Journal of Matematik (MJM)

A. B. Abdallaha, M. Dlalaa, and M. A. Hammamib, "A new lyapunov function for stability of time-varying nonlinear perturbed systems," Systems and Control Letters, vol. 56, pp. $179-187,2007$

F. Brauer, "Perturbations of nonlinear systems of differential equations," Journal of Mathematical Analysis and Apllications, vol. 37, pp. 214-222, 1972.

S. N. Chow and J. A. Yorke, "Lyapunov theory and perturbation of stable and asymptotically stable systems," Journal of Differential Equations, vol. 15, pp. 308-321, 1974.

M. Dlala and M. A. Hammami, "Uniform exponential practical stability of impulsive perturbed systems," Jour nal of Dynamical and Control Systems, vol. 13, no. 3, pp. 373-386, July 2007 .

S. G. Hristova and A. Georgieva, "Practical stability in terms of two measures for impulsive differential equations with supremum," International Journal of Differen tial Equations, vol. Volume 2011, Article ID 703189, 13 pages, 2011.

X. Liu and Q. Wang, "On stability in terms of two measures for impulsive systems of functional differential equations," Journal of Mathematical Analysis and Applications, vol. 326, pp. 252-265, 2007.

V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, vol. 6. World Scientific, 1989.

V. Lakshmikantham and X. Liu, "Stability criteria for impulsive differential equations in terms of two measures," Journal of Mathematical Analysis and Applications, vol. 137, pp. 591-604, 1988.

X. Liu and S. Sivasundaram, "Stability of nonlinear systems under constantly acting perturbations," International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, pp. 273-278, 1995.

Y. Liu and S. Zhaob, "A new approach to practical stability of impulsive functional differential equations in terms of two measures," Journal of Computational and Applied Mathematics, vol. 223, pp. 449-458, 2009.

J. Shen, "Razumikhin techniques in impulsive functional differential equations.," Nonlinear Analysis, vol. 36 , pp. 119-130, 1999 .

P. Singh, S. K.Srivastava, and K. Kaur, "Uniform practical stability in terms of two measures with effect of delay at the time of impulses," Nonlinear Engineering, vol. 5 , no. 2 , pp. 93-97, 2016 .

A. Soliman, "On stability of perturbed impulsive differential systems," Applied Mathematics and Computation, vol. 133 , pp. $105-117,2002$.

A. Sood and S. K.Srivastava, "Lyapunov Approach for Stability of Integro Differential Equations with Non Instantaneous Impulse Effect," Malaya Journal of Mathe matik, vol. 4, no. 1, pp. 119-125, 2016.

I. M. Stamova, "Razumikhin type theorems on stability in terms of two measures for impulsive functional differential systems," Note di Matematica, vol. 26, no. 2 , pp. $69-80,2006$.

I. M. Stamova, "Vector lyapunov functions for practical stability of nonlinear impulsive functional differential equations," Journal of Mathematical Analysis and Applications, vol. 325 , pp. 612-623, 2007.

I. M. Stamova and G. T. Stamov, "Lyapunov razumikhin method for impulsive functional differential equations and applications to the population dynamics," Jourmal of Computational and Applied Mathematics, vol. 130, pp. $163-171,2001$.

P. Wang and H. Lian, "On the stability in terms of two measures for perturbed impulsive integro-differential equations," Journal of Mathematical Analysis and Applications, vol. 313 , pp. 642-653, 2006.

Y. Zhang and J. Sun, "Practical stability of impulsive functional differential equations in terms of two measurements," Computers and Mathematics with Applications, vol. 48 , pp. $1549-1556,2004$.