Exponential Stability Analysis of Difference Equation for Impulsive System

Elizabeth S., Nirmal Veena S.


In this paper, we study the exponential stability of impulsive difference equations with exponential decay and the uniformity of the stability is obtained by using Lyapunov functions. Theorems on exponential and uniform exponential stability are obtained, which shows that certain impulsive perturbations may make unstable systems exponentially stable.


Difference equation; Uniform exponential stability; Lyapunov functions; Impulsive system

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R.P. Agarwal and P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers (1997).

R.P. Agarwal and D. O’Regan, Difference equations in abstract spaces, J. Austral. Math. Soc.(Series A) 64 (1998), 277 – 284.

S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer–Verlag, New York (2004).

L. Gupta aand M.M. Jin, Global asymptotic stability of discrete-time analog neural networks, IEEE Trans. Neural Netw. 7 (6) (1996), 1024 – 1031.

Z. Jiang and Y. Wang, A converse Lyapunov theorem for discrete-time systems with disturbances, Systems and Control Letters 45 (2002), 49 – 58.

C.M. Kellett and A.R. Teel, On robustness of stability and Lyapunov functions for discontinuous difference equations, in Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, December 2002.

B. Liu, X. Liu, K. Teo and Q. Wang, Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA Journal of Applied Mathematics 71 (2006), 47 – 61.

N. Linh and V. Phat, Exponential stability of nonlinear time-varying differential equations and applications, Electronic Journal of Differential Equations 2001 (34) (2001), 1 – 13.

P.M. Pardalos and V. Yatsenko, Optimization and Control of Bilinear Systems: Theory, Algorithms and Applications, Springer, Berlin (2008).

Q. Zhang and Z. Zhou, Global attractivity of a nonautonomous discrete logistic model, Hokkaido Math. J. 29 (2000), 37 – 44.

Y. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, Journal of Mathematical Analysis and Applications 279 (2) (2003), 639 – 650.

S.H. Strgate, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, Perseus Books, New York (1994).

R.E. Mickens, Difference Equations Theory, Applications and Advance Topics, 3nd ed., CRC Press, USA (2015).

Q. Wang and X. Liu, Exponential stability for impulsive delay differential equations by Razumikhin method, Journal of Mathematical Analysis and Application 309 (2005), 462 – 473.

DOI: http://dx.doi.org/10.26713%2Fjims.v9i3.933

eISSN 0975-5748; pISSN 0974-875X