Some new difference inequalities and an application to discrete-time control systems. (English) Zbl 1263.93143

Summary: Two new nonlinear difference inequalities are considered, where the inequalities consist of multiple iterated sums, and composite function of nonlinear function and unknown function may be involved in each layer. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to the stability problem of a class of linear control systems with nonlinear perturbations.


93C55 Discrete-time control/observation systems
93C73 Perturbations in control/observation systems
93C10 Nonlinear systems in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93D99 Stability of control systems
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[1] T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292-296, 1919. · JFM 47.0399.02
[2] R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643-647, 1943. · Zbl 0061.18502
[3] D. S. Mitrinović, J. E. Pe, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, The Netherlands, 1991. · Zbl 0744.26011
[4] D. Baĭnov and P. Simeonov, Integral Inequalities and Applications, vol. 57, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. · Zbl 0759.26012
[5] B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197, Academic Press, New York, USA, 1998. · Zbl 1032.26008
[6] W. Zhang and S. Deng, “Projected Gronwall-Bellman’s inequality for integrable functions,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 393-402, 2001. · Zbl 0992.26013
[7] R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599-612, 2005. · Zbl 1078.26010
[8] B.-I. Kim, “On some Gronwall type inequalities for a system integral equation,” Bulletin of the Korean Mathematical Society, vol. 42, no. 4, pp. 789-805, 2005. · Zbl 1090.26017
[9] O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 349-358, 2006. · Zbl 1103.26018
[10] W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis A, vol. 64, no. 9, pp. 2112-2128, 2006. · Zbl 1094.26011
[11] R. P. Agarwal, C. S. Ryoo, and Y.-H. Kim, “New integral inequalities for iterated integrals with applications,” Journal of Inequalities and Applications, vol. 2007, Article ID 24385, 18 pages, 2007. · Zbl 1133.26304
[12] W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 144-154, 2007. · Zbl 1193.26014
[13] R. P. Agarwal, Y.-H. Kim, and S. K. Sen, “New retarded integral inequalities with applications,” Journal of Inequalities and Applications, vol. 2008, Article ID 908784, 15 pages, 2008. · Zbl 1151.45001
[14] W.-S. Wang and C.-X. Shen, “On a generalized retarded integral inequality with two variables,” Journal of Inequalities and Applications, vol. 2008, Article ID 518646, 9 pages, 2008. · Zbl 1151.45010
[15] C.-J. Chen, W.-S. Cheung, and D. Zhao, “Gronwall-bellman-type integral inequalities and applications to BVPs,” Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009. · Zbl 1176.35007
[16] A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 7887-7899, 2011. · Zbl 1220.26012
[17] B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,” Indian Journal of Pure and Applied Mathematics, vol. 9, no. 12, pp. 1282-1290, 1978. · Zbl 0393.93018
[18] W.-S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 708-724, 2006. · Zbl 1116.26016
[19] B. G. Pachpatte, Integral and Finite Difference Inequalities and Applications, vol. 205 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1104.26015
[20] Q.-H. Ma and W.-S. Cheung, “Some new nonlinear difference inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 339-351, 2007. · Zbl 1121.26019
[21] W.-S. Wang, “A generalized sum-difference inequality and applications to partial difference equations,” Advances in Difference Equations, vol. 2008, Article ID 695495, 12 pages, 2008. · Zbl 1149.39010
[22] W.-S. Wang, “Estimation on certain nonlinear discrete inequality and applications to boundary value problem,” Advances in Difference Equations, vol. 2009, Article ID 708587, 8 pages, 2009. · Zbl 1168.26318
[23] X.-M. Zhang and Q.-L. Han, “Delay-dependent robust H\infty filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 12, pp. 1466-1470, 2006.
[24] K. L. Zheng, S. M. Zhong, and M. Ye, “Discrete nonlinear inequalities in time control systems,” in Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA ’09), pp. 403-406, 2009.
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