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de la Sen, M.

Asymptotic comparison of the solutions of linear time-delay systems with point and distributed lags with those of their limiting equations. (English) Zbl 1191.34096

Abstr. Appl. Anal. 2009, Article ID 216746, 37 p. (2009).
This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.
Reviewer: Shangjiang Guo (Hunan)

Cited in 3 Documents

MSC:

34K25 Asymptotic theory of functional-differential equations
34K06 Linear functional-differential equations
34K20 Stability theory of functional-differential equations
34K27 Perturbations of functional-differential equations

Keywords:

functional differential equation; Volterra class dynamics; Perron-type theorem; global stability
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References:

[1] M. De la Sen and N. Luo, “Discretization and FIR filtering of continuous linear systems with internal and external point delays,” International Journal of Control, vol. 60, no. 6, pp. 1223-1246, 1994. · Zbl 0813.93013 · doi:10.1080/00207179408921518
[2] T. A. Burton, Stability and Periodic Solutions of Ordinary and Differential Equations, vol. 178 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1985. · Zbl 0635.34001
[3] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[4] S.-I. Niculescu, Delay Effects on Stability. A Robust Control Approach, vol. 269 of Lecture Notes in Control and Information Sciences, Edited by M. Thoma, M. Morari, Springer, Berlin, Germany, 2001. · Zbl 0997.93001
[5] M. De la Sen and N. Luo, “On the uniform exponential stability of a wide class of linear time-delay systems,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 456-476, 2004. · Zbl 1046.34086 · doi:10.1016/j.jmaa.2003.08.048
[6] M. De la Sen, “Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 621-650, 2006. · Zbl 1111.93072 · doi:10.1016/j.jmaa.2005.08.038
[7] R. F. Datko, “Time-delayed perturbations and robust stability,” in Differential Equations, Dynamical Systems and Control Science, vol. 152 of Lecture Notes in Pure ands Applied Mathematics, pp. 457-468, Marcel Dekker, New York, NY, USA, 1994. · Zbl 0792.93098
[8] C. F. Alastruey, M. De la Sen, and J. R. González de Mendívil, “The stabilizability of integro-differential systems with two distributed delays,” Mathematical and Computer Modelling, vol. 21, no. 8, pp. 85-94, 1995. · Zbl 0826.45005 · doi:10.1016/0895-7177(95)00041-Y
[9] N. U. Ahmed, “Optimal control of infinite-dimensional systems governed by integro differential equations,” in Differential Equations, Dynamical Systems and Control Science, vol. 152 of Lecture Notes in Pure and Applied Mathematics, pp. 383-402, Marcel Dekker, New York, NY, USA, 1994. · Zbl 0794.49002
[10] M. R. S. Kulenović, G. Ladas, and A. Meimaridou, “Stability of solutions of linear delay differential equations,” Proceedings of the American Mathematical Society, vol. 100, no. 3, pp. 433-441, 1987. · Zbl 0645.34061 · doi:10.2307/2046425
[11] M. De la Sen, “On impulsive time-varying systems with unbounded time-varying point delays: stability and compactness of the relevant operators mapping the input space into the state and output spaces,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 1, pp. 79-129, 2007. · Zbl 1160.34071 · doi:10.1216/rmjm/1181069321
[12] Y. Jiang and Y. Jurang, “Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses,” Journal of Mathematical Analysis and Applications, vol. 207, no. 2, pp. 388-396, 1997. · Zbl 0877.34054 · doi:10.1006/jmaa.1997.5276
[13] T. Sengadir, “Asymptotic stability of nonlinear functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications. Series A, vol. 28, no. 12, pp. 1997-2003, 1997. · Zbl 0869.34060 · doi:10.1016/S0362-546X(96)00029-6
[14] M. De la Sen, “An algebraic method for pole placement in multivariable systems with internal and external point delays by using single rate or multirate sampling,” Dynamics and Control, vol. 10, no. 1, pp. 5-31, 2000. · Zbl 0953.93036 · doi:10.1023/A:1008351026971
[15] M. De la Sen, “On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 382-401, 2007. · Zbl 1117.93034 · doi:10.1016/j.amc.2007.01.053
[16] F. Chen and C. Shi, “Global attractivity in an almost periodic multi-species nonlinear ecological model,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 376-392, 2006. · Zbl 1099.92069 · doi:10.1016/j.amc.2005.12.024
[17] X. Meng, W. Xu, and L. Chen, “Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 365-378, 2007. · Zbl 1113.92070 · doi:10.1016/j.amc.2006.09.133
[18] L. White, F. White, Y. Luo, and T. Xu, “Estimation of parameters in carbon sequestration models from net ecosystem exchange data,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 864-879, 2006. · Zbl 1101.92059 · doi:10.1016/j.amc.2006.02.014
[19] R. R. Sarkar, B. Mukhopadhyay, R. Bhattacharyya, and S. Banerjee, “Time lags can control algal bloom in two harmful phytoplankton-zooplankton system,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 445-459, 2007. · Zbl 1111.92065 · doi:10.1016/j.amc.2006.07.113
[20] L. Bakule and J. M. Rossell, “Overlapping controllers for uncertain delay continuous-time systems,” Kybernetika, vol. 44, no. 1, pp. 17-34, 2008. · Zbl 1146.93004
[21] X. Xiang and N. U. Ahmed, “Existence of periodic solutions of semilinear evolution equations with time lags,” Nonlinear Analysis: Theory, Methods & Applications. Series A, vol. 18, no. 11, pp. 1063-1070, 1992. · Zbl 0765.34057 · doi:10.1016/0362-546X(92)90195-K
[22] J. Wang, X. Xiang, and W. Wei, “Linear impulsive periodic system with time-varying generating operators on Banach space,” Advances in Difference Equations, vol. 2007, Article ID 26196, 16 pages, 2007. · Zbl 1148.39015 · doi:10.1155/2007/26196
[23] J. Wang, X. Xiang, W. Wei, and Q. Chen, “Existence and global asymptotical stability of periodic solution for the T-periodic logistic system with time-varying generating operators and T0-periodic impulsive perturbations on Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 524945, 16 pages, 2008. · Zbl 1154.34030 · doi:10.1155/2008/524945
[24] J. Wang, X. Xiang, and W. Wei, “Periodic solutions of semilinear impulsive periodic system with time-varying generating operators on Banach space,” Mathematical Problems in Engineering, vol. 2008, Article ID 183489, 15 pages, 2008. · Zbl 1175.34055 · doi:10.1155/2008/183489
[25] B. Sasu, “Robust stability and stability radius for variational control systems,” Abstract and Applied Analysis, vol. 2008, Article ID 381791, 29 pages, 2008. · Zbl 1395.93469 · doi:10.1155/2008/381791
[26] G. Grammel, “Homogenization of elliptic differential equations in one-dimensional spaces,” Abstract and Applied Analysis, vol. 2007, Article ID 98538, 6 pages, 2007. · Zbl 1153.34317 · doi:10.1155/2007/98538
[27] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, Mass, USA, 1965. · Zbl 0154.09301
[28] M. Pituk, “Asymptotic behavior and oscillation of functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1140-1158, 2006. · Zbl 1113.34059 · doi:10.1016/j.jmaa.2005.09.081
[29] M. Pituk, “A Perron type theorem for functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 24-41, 2006. · Zbl 1102.34060 · doi:10.1016/j.jmaa.2005.04.027
[30] G. Da Prato and M. Iannelli, “Linear integro-differential equations in Banach spaces,” Rendiconti del Seminario Matematico dell’Università di Padova, vol. 62, pp. 207-219, 1980. · Zbl 0451.45014
[31] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, New York, NY, USA, 1958.
[32] S. Barnett, Polynomials and Linear Control Systems, vol. 77 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1983. · Zbl 0528.93003
[33] U. Mackenroth, Robust Control Systems. Theory and Case Studies, Springer, Berlin, Germany, 2004. · Zbl 1069.93001
[34] D. S. Watkins, The Matrix Eigenvalue Problem. GR and Krylov Subspace Methods, SIAM, Philadelphia, Pa, USA, 2007. · Zbl 1142.65038 · doi:10.1137/1.9780898717808
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