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New conditions for the exponential stability of pseudolinear difference equations in Banach spaces. (English) Zbl 1470.39036

Summary: We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

MSC:

39A30 Stability theory for difference equations
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[1] Agarwal, R. P.; Thompson, H. B.; Tisdell, C. C., Difference equations in Banach spaces, Computers & Mathematics with Applications, 45, 6–9, 1437-1444, (2003) · Zbl 1057.39007 · doi:10.1016/s0898-1221(03)00100-7
[2] Gil’, M. I.; Medina, R., The freezing method for linear difference equations, Journal of Difference Equations and Applications, 8, 5, 485-494, (2002) · Zbl 1005.39005 · doi:10.1080/10236190290017478
[3] Lee, S. M.; Park, J. H., Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities, Applied Mathematics and Computation, 200, 1, 429-436, (2008) · Zbl 1146.93017 · doi:10.1016/j.amc.2007.11.031
[4] Memarbashi, R., Sufficient conditions for the exponential stability of non-autonomous difference equations, Applied Mathematics Letters, 21, 3, 232-235, (2008) · Zbl 1171.39002 · doi:10.1016/j.aml.2007.03.014
[5] Medina, R., Exponential stabilization of nonlinear discrete-time systems, Journal of Difference Equations and Applications, 17, 5, 697-708, (2011) · Zbl 1221.93245 · doi:10.1080/10236190903168007
[6] Sasu, B.; Sasu, A. L., Stability and stabilizability for linear systems of difference equations, Journal of Difference Equations and Applications, 10, 12, 1085-1105, (2004) · Zbl 1064.39011 · doi:10.1080/10236190412331314178
[7] Hsien, T.-L.; Lee, C.-H., Exponential stability of discrete time uncertain systems with time-varying delay, Journal of the Franklin Institute, 332, 4, 479-489, (1995) · Zbl 0853.93085 · doi:10.1016/0016-0032(95)00058-5
[8] Bay, N. S.; Phat, V. N., Stability analysis of nonlinear retarded difference equations in Banach spaces, Computers & Mathematics with Applications, 45, 6–9, 951-960, (2003) · Zbl 1053.39004 · doi:10.1016/s0898-1221(03)00068-3
[9] Medina, R.; Gil, M. I., The freezing method for abstract nonlinear difference equations, Journal of Mathematical Analysis and Applications, 330, 1, 195-206, (2007) · Zbl 1115.39010 · doi:10.1016/j.jmaa.2006.07.074
[10] Ngoc, P. H. A.; Hieu, L. T., New criteria for exponential stability of nonlinear difference systems with time-varying delay, International Journal of Control, 86, 9, 1646-1651, (2013) · Zbl 1278.93218 · doi:10.1080/00207179.2013.792004
[11] Peuteman, J.; Aeyels, D., Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying, SIAM Journal on Control and Optimization, 37, 4, 997-1010, (1999) · Zbl 0924.34047 · doi:10.1137/s0363012997323862
[12] Solo, V., On the stability of slowly time-varying linear systems, Mathematics of Control, Signals, and Systems, 7, 4, 331-350, (1994) · Zbl 0833.93047 · doi:10.1007/BF01211523
[13] Sundarapandian, V., Exponential stabilizability and robustness analysis for discrete-time nonlinear systems, Applied Mathematics Letters. An International Journal of Rapid Publication, 18, 7, 757-764, (2005) · Zbl 1122.93054 · doi:10.1016/j.aml.2004.05.016
[14] Wang, J.; Li, X., Improved global exponential stability for delay difference equations with impulses, Applied Mathematics and Computation, 217, 5, 1933-1938, (2010) · Zbl 1217.39026 · doi:10.1016/j.amc.2010.06.048
[15] Banks, S. P.; Moser, A.; Mccaffrey, D., Robust exponential stability and evolution equations, Archives of Control Sciences, 4, 3-4, 261-279, (1995) · Zbl 0848.93051
[16] Dvirnyi, A. I.; Slyn’ko, V. I., On stability of solutions of nonlinear nonstationary systems of impulsive differential equations in a critical case, Nonlinear Oscillations, 14, 4, 472-496, (2012) · Zbl 1334.34038 · doi:10.1007/s11072-012-0171-7
[17] Dvirnyi, A. I.; Slyn’ko, V. I., Global stability of solutions of nonstationary monotone differential equations with impulsive action in the pseudolinear form, Nonlinear Oscillations, 14, 2, 193-210, (2011) · Zbl 1334.34137 · doi:10.1007/s11072-011-0151-3
[18] Hong, K. S.; Wu, J. W.; Lee, K.-I., New conditions for the exponential stability of evolution equations, IEEE Transactions on Automatic Control, 39, 7, 1432-1436, (1994) · Zbl 0825.93678 · doi:10.1109/9.299627
[19] Martynyuk, A. A., Novel bounds for solutions of nonlinear differential equations, Applied Mathematics, 6, 1, 182-194, (2015) · doi:10.4236/am.2015.61018
[20] Medina, R., Exponential stability of slowly varying discrete systems with multiple state delays, International Journal of Robust and Nonlinear Control, 23, 13, 1496-1509, (2013) · Zbl 1278.93216 · doi:10.1002/rnc.2835
[21] Bylov, B. F.; Grobman, B. M.; Nemickii, V. V.; Vinograd, R. E., Theory of Lyapunov Exponents, Moscow, Russia: Nauka, Moscow, Russia · Zbl 0144.10702
[22] Gil’, M. I., Norm Estimates for Operator-Valued Functions and Applications, (1995), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0840.47006
[23] Medina, R.; Martinez, C., Exponential stability criteria for discrete time-delay systems, International Journal of Robust and Nonlinear Control, 25, 4, 527-541, (2015) · Zbl 1312.93087 · doi:10.1002/rnc.3103
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