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http://dx.doi.org/10.4134/JKMS.j150428

A CONVERSE THEOREM ON h-STABILITY VIA IMPULSIVE VARIATIONAL SYSTEMS  

Choi, Sung Kyu (Department of Mathematics Chungnam National University)
Koo, Namjip (Department of Mathematics Chungnam National University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1115-1131 More about this Journal
Abstract
In this paper we develop useful relations which estimate the difference between the solutions of nonlinear impulsive differential systems with different initial values. Then we obtain the converse h-stability theorem of Massera's type for the nonlinear impulsive systems by employing the $t_{\infty}$-similarity of the associated impulsive variational systems and relations.
Keywords
impulsive variational systems; h-stability; impulsive differential systems; $t_{\infty}$-similarity; converse stability theorem;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 D. D. Bainov, S. I. Kostadinov, and A. D. Myshkis, Kinematical similarity and exponential dichotomy of linear abstract impulsive differential equations, Internat. J. Theoret. Phys. 33 (1994), no. 2, 487-497.   DOI
2 D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Limited, Chichester, 1989.
3 D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
4 S. K. Choi and N. Koo, Variationally stable impulsive differential systems, Dyn. Syst. 30 (2015), no. 4, 435-449.   DOI
5 S. K. Choi, N. Koo, and C. Ryu, h-Stability of linear impulsive differential equations via similarity, J. Chungcheong Math. Soc. 24 (2011), 393-400.
6 S. K. Choi, N. Koo, and C. Ryu, Stability of linear impulsive differential equations via $t_{\infty}$-similarity, J. Chung- cheong Math. Soc. 26 (2013), 811-819.   DOI
7 S. K. Choi, N. Koo, and H. S. Ryu, h-Stability of differential systems via $t_{\infty}$-similarity, Bull. Korean Math. Soc. 34 (1997), no. 3, 371-383.
8 R. Conti, Sulla $t_{\infty}$-similitudine tra matrici e la stabilita dei sistemi differenzialelineari, Atti. Acc. Naz. Lincei, Rend. Cl. Fis. Mat. Nat. (8) 19 (1955), 247-250.
9 F. M. Dannan and S. Elaydi, Lipschitz stability of nonlinear systems of differential equations, J. Math. Anal. Appl. 113 (1986), no. 2, 562-577.   DOI
10 J. V. Devi, A variation of the Lyapunov second method to impulsive differential equations, J. Math. Anal. Appl. 177 (1993), no. 1, 190-200.   DOI
11 M. Dlala and M. A. Hammami, Uniform exponential practical stability of impulsive perturbed systems, J. Dyn. Control Syst. 13 (2007), no. 3, 373-386.   DOI
12 G. K. Kulev and D. D. Bainov, Lipschitz stability of impulsive systems of differential equations, Internat. J. Theoret. Phys. 30 (1991), no. 5, 737-756.   DOI
13 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Co. Pte. Ltd., Teaneck, NJ, 1989.
14 V. Lakshmikantham and S. G. Deo, Method of Variation of Parameters for Dynamical Systems, Gordon and Breach Science Publishers, Amsterdam, 1998.
15 V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol I and II, Academic Press, New York, 1969.
16 R. Medina and M. Pinto, Uniform asymptotic stability of solutions of impulsive differential equations, Dynam. Systems Appl. 5 (1996), no. 1, 97-107.
17 M. Pinto, Perturbations of asymptotically stable differential systems, Analysis 4 (1984), no. 1-2, 161-175.
18 M. Pinto, Stability of nonlinear differential systems, Appl. Anal. 43 (1992), no. 1-2, 1-20.   DOI
19 P. S. Simeonov and D. D. Bainov, Stability of the solutions of singularly perturbed systems with impulse effect, J. Math. Anal. Appl. 136 (1988), no. 2, 575-588.   DOI
20 P. S. Simeonov and D. D. Bainov, Exponential stability of the solutions of singularly perturbed systems with impulse effect, J. Math. Anal. Appl. 151 (1990), no. 2, 462-487.   DOI
21 T. Yoshizawa, Stability Theory by Liapunov's Second Method, The Mathematical Society of Japan, Tokyo, 1966.
22 J. Shen and X. Liu, Global existence results for impulsive differential equations, J. Math. Anal. Appl. 314 (2006), no. 2, 546-557.   DOI