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Impulsive effects on global stability of models based on impulsive differential equations with “supremum” and variable impulsive perturbations. (English) Zbl 1284.34020

Summary: Sufficient conditions are investigated for the global stability of the solutions to models based on nonlinear impulsive differential equations with “supremum” and variable impulsive perturbations. The main tools are the Lyapunov functions and Razumikhin technique. Two illustrative examples are given to demonstrate the effectiveness of the obtained results.

MSC:

34A37 Ordinary differential equations with impulses
34D20 Stability of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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[1] Alzabut, J. O., Stamov, G. T., and Sermutlu, E. On almost periodic solutions for an impulsive delay logarithmic population model. Mathematical and Computer Modelling, 51, 625-631 (2010) · Zbl 1190.34087 · doi:10.1016/j.mcm.2009.11.001
[2] Lakshmikantham, V., Bainov, D. D., and Simeonov, P. S. Theory of Impulsive Differential Equations, World Scientific, New Jersey (1989) · Zbl 0719.34002 · doi:10.1142/0906
[3] Samoilenko, A. M. and Perestyuk, N. A. Differential Equations with Impulse Effect (in Russian), Visca Skola, Kiev (1987)
[4] Stamov, G. T. Almost periodic models of impulsive Hopfield neural networks. Journal of Mathematics of Kyoto University, 49, 57-67 (2009) · Zbl 1177.34058
[5] Stamov, G. T. On the existence of almost periodic solutions for impulsive Lasota-Wazevska model. Applied Mathematics Letters, 22, 516-520 (2009) · Zbl 1179.34093 · doi:10.1016/j.aml.2008.07.002
[6] Popov, E. R. Automatic Regulation and Control (in Russian), 2nd ed., Nauka, Moscow (1988)
[7] Stamova, I. M., Stamov, T., and Simeonova, N. Impulsive control on global exponential stability for cellular neural networks with supremums. Journal of Vibration and Control, 19, 483-490 (2013) · Zbl 1348.93150 · doi:10.1177/1077546312441042
[8] Hale, J. K. Theory of Functional Differential Equations, Springer-Verlag, Heidelberg, 130-172 (1977) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2
[9] Sun, J. and Wu, Q. Impulsive control for the stabilization and synchronization of Lur’e systems. Applied Mathematics and Mechanics (English Edition), 25, 322-328 (2004) DOI 10.1007/BF02437335 · Zbl 1151.93365 · doi:10.1007/BF02437335
[10] Zhi, L. and Yan, L. Asymptotic stability for impulsive functional differential equations. Applied Mathematics and Mechanics (English Edition), 30, 1317-1324 (2009) DOI 10.1007/s10483-009-1011-z · Zbl 1192.34095 · doi:10.1007/s10483-009-1011-z
[11] Bainov, D. D., Domshlak, Y., and Milusheva, S. Partial averaging for impulsive differential equations with supremum. Georgian Mathematical Journal, 3, 11-26 (1996) · Zbl 0837.34018 · doi:10.1007/BF02256795
[12] Caballero, J., Lopez, B., and Sadarangani, K. On monotonic solutions of an integral equation of Volterra type with supremum. Journal of Mathematical Analysis and Applications, 305, 304-315 (2005) · Zbl 1076.45002 · doi:10.1016/j.jmaa.2004.11.054
[13] He, Z. M., Wang, P. G., and Ge, W. G. Periodic boundary value problem for first order impulsive differential equations with supremum. Indian Journal of Pure and Applied Mathematics, 34, 133-143 (2003) · Zbl 1029.34067
[14] Milusheva, S. and Bainov, D. D. Averaging method for neutral type impulsive differential equations with supremums. Annales de la Faculte des Sciences de Toulouse, 12(3), 391-403 (1991) · Zbl 0760.34039 · doi:10.5802/afst.733
[15] Stamova, I. M. Lyapunov-Razumikhin method for impulsive differential equations with “supremum”. IMA Journal of Applied Mathematics, 76, 573-581 (2011) · Zbl 1222.93199 · doi:10.1093/imamat/hxq055
[16] Bainov, D. D. and Dishliev, A. B. The phenomenon “beating” of the solutions of impulsive functional differential equations. Communications in Applied Analysis, 1, 435-441 (1997) · Zbl 0896.34011
[17] Stamova, I. M. Boundedness of impulsive functional differential equations with variable impulsive perturbations. Bulletin of the Australian Mathematical Society, 77, 331-345 (2008) · Zbl 1162.34066 · doi:10.1017/S0004972708000439
[18] Simeonov, P. S. and Bainov, D. D. Stability with respect to part of the variables in systems with impulse effect. Journal of Mathematical Analysis and Applications, 117, 247-263 (1986) · Zbl 0588.34044 · doi:10.1016/0022-247X(86)90259-3
[19] Razumikhin, B. S. Stability of Systems with Retardation (in Russian), Nauka, Moscow (1988)
[20] Stamova, I. M. Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter, New York (2009) · Zbl 1189.34001 · doi:10.1515/9783110221824
[21] Wang, Q. and Liu, X. Impulsive stabilization of delay differential system via the Lyapunov-Razumikhin method. Applied Mathematics Letters, 20, 839-845 (2007) · Zbl 1159.34347 · doi:10.1016/j.aml.2006.08.016
[22] Lakshmikantham, V., Leela, S., and Martynyuk, A. A. Practical Stability of Nonlinear Systems, World Scientific, New Jersey, 189-199 (1990) · Zbl 0753.34037 · doi:10.1142/1192
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