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Impulsive stabilization of functional differential equations via Lyapunov functionals. (English) Zbl 0955.34069

The authors discuss the use of the second Lyapunov method and the use of Lyapunov functionals in the analysis of impulsive stabilization problems concerning functional-differential equations of the form \[ x'(t)= f(t, x_t),\quad t\geq t_0,\quad \Delta_x= I_k(t, x(t^-)),\quad t= t_k,\quad k\in \mathbb{Z}^+. \]

MSC:

34K45 Functional-differential equations with impulses
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[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Bainov, D. D.; Simeonov, P. S., Systems with Impulse Effect: Stability Theory and Applications (1989), Horwood: Horwood Chicester · Zbl 0676.34035
[3] Lakshmikantham, V.; Liu, X., On quasi stability for impulsive differential systems, Nonlinear Anal., 13, 819-828 (1989) · Zbl 0688.34032
[4] Gopalsamy, K.; Zhang, B. G., On delay differential equations with impulses, J. Math. Anal. Appl., 139, 110-122 (1989) · Zbl 0687.34065
[5] Bainov, D. D.; Covachev, V.; Stamova, I., Estimates of the solutions of impulsive quasi-linear functional differential equations, Ann. Fac. Sci. Toulouse, 12, 149-161 (1991) · Zbl 0749.34039
[6] Bainov, D. D.; Covachev, V.; Stamova, I., Stability under persistent disturbances of impulsive differential-difference equations of neutral type, J. Math. Anal. Appl., 187, 790-808 (1994) · Zbl 0811.34057
[7] Bainov, D. D.; Kulev, G.; Stamova, I., Global stability of the solutions of impulsive differential difference equations, SUT J. Math., 31, 55-71 (1995) · Zbl 0833.34070
[8] Anokhin, A. V.; Berezansky, L.; Braverman, E., Exponential stability of linear delay impulsive differential equations, J. Math. Anal. Appl., 193, 923-941 (1995) · Zbl 0837.34076
[9] Yu, J. S.; Zhang, B. G., Stability theorems for delay differential equations with impulses, J. Math. Anal. Appl., 198, 197-285 (1996) · Zbl 0853.34068
[10] Liu, X. Z., Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10, 11-19 (1993) · Zbl 0789.93101
[11] Yu, J., On the stability of impulsive delay differential equations caused by impulses, Acta. Math. Sinica (N. S.), 13, 45-52 (1997)
[12] Shen, J.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear Anal., 33, 519-531 (1998) · Zbl 0933.34083
[13] Shen, J., Some asymptotic stability results of impulsive integro-differential equations, Chinese Ann. Math. Ser. A, 17, 165-759 (1996) · Zbl 0877.34051
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