Abstract
Problems related to the stability and asymptotic stability of sets of sufficiently general type with respect to impulsive systems are considered. The research is done by means of piecewise continuous anxiliary functions which are an analogue of the classical Lyapunov functions. It is proved that the existence of such functions with certain properties is a sufficient condition for various types of stability and asymptotic stability of sets with respect to impulsive systems.
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References
Dishliev, A. B., and Bainov, D. D. (1984). Sufficient conditions for absence of “beating” in systems of differential equations with impulses,Applicable Analysis,18, 67–73.
Dishliev, A. B., and Bainov, D. D. (1985). Conditions for the absence of the phenomenon “beating” for systems of impulse differential equations,Bulletin of the Institute of Mathematics Academia Sinica,13(3), 237–256.
Hale, J. K., and Stokes, A. P. (1960). Behaviour of solutions near integral manifolds,Archive of Rational Mechanics and Analysis,6, 133–170.
Leela, S. (1977). Stability of differential systems with impulsive perturbations in terms of two measures,Nonlinear Analysis, Theory, Methods and Applications,1(6), 667–677.
Lefschetz, S. (1958). Liapunov and stability in dynamical systems,Boletin Sociedad Matematica Mexicana,3, 25–39.
Mil'man, V. D., and Myshkis, A. D. (1960). On the stability of motion in the presence of impulses,Siberian Mathematical Journal, 233–237 (in Russian).
Pandit, S. G. (1977). On the stability of impulsively perturbed differential systems,Bulletin of the Australian Mathematical Society,17, 423–432.
Pavlidis, T. (1967). Stability of systems described by differential equations containing impulses,IEEE Transactions AC-12, 43–45.
Rama Mohana Rao, and Sree Hari Rao, V. (1977). Stability of impulsively perturbed systems,Bulletin of the Australian Mathematical Society,16, 99–110.
Rouche, N., Habets, P., and Laloy, M. (1977).Stability theory by Liapunowv's direct method, Springer-Verlag, New York.
Samoilenko, A. M., and Perestyuk, N. A. (1981) On the stability of the solutions of systems with impulse effect,Differential'nye Uravn. 11, 1995–2001 (in Russian).
Simeonov, P. S., and Bainov, D. D. (1985a). The second method of Liapunov for systems with impulse effect,Tamkang Journal of Mathematics,16(4), 19–40.
Simeonov, P. S., and Bainov, D. D. (1985b). Stability under persistent disturbances for systems with impulse effect,Journal of Mathematical Analysis and Applications,109(2), 546–563.
Simeonov, P. S., and Bainov, D. D. (1986). Stability with respect to part of the variables in systems with impulse effect,Journal of Mathematical Analysis and Applications,117(1), 247–263.
Yoshizawa, T. (1954). Liapunov's function and boundedness of solutions,Funkcialaj Ekvacioj,2, 95–142.
Yoshizawa, T. (1962). Asymptotic behavior of solutions of non-autonomous system near sets,Journal of Mathematics Kyoto Universiety,1, 303–323.
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Kulev, G.K., Bainov, D.D. Stability of sets for impulsive systems. Int J Theor Phys 28, 195–207 (1989). https://doi.org/10.1007/BF00669811
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DOI: https://doi.org/10.1007/BF00669811