Exponential stability of a state-dependent delay system

Discrete and Continuous Dynamical Systems

Volumn 18, Issue 4, 2007, Pages 773-791

  Gyori, István ^a   Hartung, Ferenc ^a  

^a UNIVERSITY OF PANNONIA   (Hungary)

Author keywords

Exponential stability; Order of exponential stability; State dependent delay; Threshold delay

Indexed keywords

EID: 35248831635     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2007.18.773     Document Type: Article

References (30)

1. L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314 (2006), 391-411.
2. K. L. Cooke, Functional differential equations: some models and perturbation problems, in "Differential Equations and Dynamical Systems" (eds. J. K. Hale and J. P. LaSalle), Academic, New York, 1967.
3. K. L. Cooke, Asymptotic theory for the delay-differential equation u′(t) = -au(t - r(u(t))), J. Math. Anal. Appl., 19 (1967), 160-173.
4. K. L. Cooke, Asymptotic equivalence of an ordinary and a functional differential equation, J. Math. Anal. Appl., 51 (1975), 187-207.
5. K. L. Cooke and W. Huang, A theorem of George Seifert and an equation with state-dependent delay, in "Delay and Differential Equations" (eds. Fink, A.M., et al.), pp. 65-77, World Scientific, Singapore, 1992.
6. K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc Amer. Math. Soc., 124 (1996),1417-1426.
7. R. D. Driver, Existence theory for a delay-differential system, Contributions to Differential Equations, 1 (1961), 317-336.
8. J. A. Gatica and J. Rivero, Qualitative behavior of solutions of some state-dependent delay equations, in "Delay and Differential Equations" (Ames, IA, 1991), 36-55, World Sci. Publishing, River Edge, NJ, 1992.
9. J. A. Gatica and P. Waltman, Existence and uniqueness of solutions of a functional-differential equation modeling thresholds, Nonlinear Anal., 8 (1984), 1215-1222.
10. J. A. Gatica and P. Waltman, A system of functional-differential equations modeling threshold phenomena, Appl. Anal., 28 (1988), 39-50.
11. I. Gyori and F. Hartung, On the exponential stability of a state-dependent delay equation, Acta Sci. Math. (Szeged), 66 (2000), 71-84.
12. I. Gyori and F. Hartung, On equi-stability with respect to parameters in functional differential equations, Nonlinear Funct. Anal. Appl., 7 (2002), 329-351.
13. I. Gyori and F. Hartung, Fundamental solution and asymptotic stability of delay differential equations, Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 261-287.
14. I. Gyori, F. Hartung and J. Turi, Numerical approximations for a class of differential equations with time- and state-dependent delays, Appl. Math. Lett., 8 (1995), 19-24.
15. J. K. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer, New York 1993.
16. T. Hara and J. Sugie, Stability region for systems of differential-difference equations, Funkcial. Ekvac., 39 (1996), 69-86.
17. T. Hara, R. Miyazaki and T. Morii, Asymptotic stability conditions for linear differential-difference equations with N delays, Dynam. Systems Appl., 6 (1997), 493-505.
18. F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Appl. Math., 174 (2005), 201-211.
19. F. Hartung and J. Turi, Stability in a class of functional differential equations with state-dependent delays, in "Qualitative Problems for Differential Equations and Control Theory" (ed. C. Corduneanu), Word Scientific, 1995, 15-31.
20. F. Hartung, J. Turi, Linearized stability in functional-differential equations with state-dependent delays, Proceedings of the conference Dynamical Systems and Differential Equations, added volume of Discrete and Continuous Dynamical Systems, (2000), 416-425.
21. F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations" (eds. A. Cañada, P. Drábek and A. Fonda), volume 3, Elsevier, North-Holand, 2006, 435-545.
22. T. Krisztin, On stability properties for one-dimensional functional differential equations, Funkcial. Ekvac., 34 (1991), 241-256.
23. Y. Kuang, 3/2 stability results for nonautonomous state-dependent delay differential equations, in "Differential Equations to Biology and to Industry" (eds. Martelli, M., et al.), pp. 261-269, World Scientific, Singapore, 1996.
24. T. Luzyanina, K. Engelborghs, and D. Roose, Numerical bifurcation analysis of differential equations with state-dependent delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 737-753.
25. R. Miyazaki, Characteristic equation and asymptotic behavior of delay-differential equation, Funkcial. Ekvac., 40 (1997), 471-481.
26. S. Sakata, Asymptotic stability for a linear system of differential-difference equations, Funkcial. Ekvac., 41 (1998), 435-449.
27. H. L. Smith, Structured population models, threshold-type delay equations and functional-differential equations, in "Delay and Differential Equations" (Ames, IA, 1991), 57-64, World Sci. Publishing, River Edge, NJ, 1992.
28. J. W.-H. So, X. Tang and X. Zou, Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.
29. P. Waltman, "Deterministic threshold models in the theory of epidemics," Lecture Notes in Biomathematics, 1 Springer-Verlag 1974.
30. T. Yoneyama, Uniform stability for n-dimensional delay-differential equations, Funkcial. Ekvac., 34 (1991), 495-504.