Robustness of asymptotic stability to small time delays

Discrete and Continuous Dynamical Systems

Volumn 13, Issue 4, 2005, Pages 1007-1034

  Li, Desheng ^a,b   Kloeden, Peter E ^c  

^a TIANJIN UNIVERSITY   (China)

^b Lanzhou Teachers College   (China)

^c GOETHE UNIVERSITY   (Germany)

Author keywords

Asymptotic stability; Continuous convergence; Delay differential equations; Differential inclusions; Exponential dissipativity; Global attractors; Lyapunov functions; Setvalued dynamical systems

Indexed keywords

EID: 28444491086     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2005.13.1007     Document Type: Article

References (40)

1. O. Arino and M. Pituk (2001), More on linear differential systems with small delays, J. Diff. Eqns 170, 381-407.
2. J.P. Aubin and A. Cellina (1984), Differential Inclusions. Springer-Verlag.
3. T. Caraballo, Nonlinear partial functional differential equations: existence and stability (2001), J. Math. Anal Applns. 262, 87-111.
4. T. Caraballo, P.E. Kloeden and J. Langa (2003), Atractores globales para sistemas diferenciales no autónomes, Cubo Matemática Educacional 5, 305-329.
5. T. Caraballo, J.A. Langa and J.C. Robinson (2001), Attractors for differential equations with variable delays, J. Math.Anal. Applns. 260, 421-438.
6. C. Chicone (2003), Inertial and slow manifolds for delay equations with small delays, J. Diff. Eqns.190, 364-406.
7. F.H. Clarke and Yu. S. Ledyaev and R. J. Stern (1998), Asymptotic stability and smooth Lyapunov functions, J. Diff. Eqns. 149, 69-114.
8. R. Datko (1994), Time delay perturbations and robust stability, in Differential Equations, Dynamical Systems and Control Science, 457-468. Lecture Notes in Pure and Appl. Math., 152, Dekker, New Yor Marcel Dekker.
9. K. Deimling (1992), Multivalued Differential Equations. De Gruyter.
10. R.D. Driver (1976), Linear differential systems with small delays, J. Diff. Eqns21, 149-167.
11. F.M. Guo, W. Yu and F.L. Huang (2003), Robustness with respect to small delays for exponential stability of non-autonomous systems, J. Math. Anal. Applns. 288, 671-679.
12. C.R. Friesecke (1993), Convergence to equilibrium for delay-diffusion equations with small delay, J. Dyn. Diff. Eqns 5, 89-103.
13. I. Gyöni and M. Pituk (2001), Special solutions of neutral functions differential equations, J. Inequal. Applns. 6, 99-117.
14. S.F. Hafstein (2004), A constructive converse Lyapunov theorem on exponential stability Discrete Cont. Dyn. Systems, Series A 10, 657-678.
15. J.K. Hale (1988), Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc.
16. J.K. Hale, (1977), Theory of Functional Differential Equations. Springer-Verlag.
17. J.K. Hale and S.M. Verduyn Lunel (1993), Introduction to Functional Differential Equations. Springer-Verlag.
18. J.K. Hale and S.M. Verduyn Lunel (2001), Effects of small delays on stability and control. Operator Theory and Analysis (Amsterdam, 1997), 275-301. Operator Theory Adv.Appl., 122, Birkhäuser.
19. V.L. Kharitonov (1999), Robust stability analysis of time delay systems: A Survey. Ann. Rev. Control 23, 185-196.
20. P.E. Kloeden (1978), General control systems, Mathematical Control Theory (W.A. Coppel, ed.), Lecture Notes in Mathematics 680, Springer-Verlag, 119-138.
21. V.B. Kolmanovskii and V.R. Nosoc (1986), Stability of Functional Differential Equations. Academic Press
22. S. Krisztin (2003), A local unstable manifold for differential equations with state-dependent delay, Discrete Cont. Dyn. Systems 9, 993-1028.
23. J. Kurzweil (1971), Small delays don't matter, in "Lecture Notes in Math." Vol. 206, Springer-Verlag, New York.
24. D.S. Li (2001), On dynamical stability in general dynamical systems, J. Math. Anal. Applns. 263, 455-478.
25. D.S. Li and X.X. Zhang (2002), On the stability in general dynamical systems and differential inclusions, J. Math. Anal. Applns. 274, 705-724.
26. W.J. Liu (2002), Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Cont. Dyn. Systems, Series B 2, 47-56.
27. H. Logemann, R. Rebarber and G. Weiss, (1996), Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM J. Control Optim. 37, 572-600.
28. S. Luckhaus (1986), Global boundedness for a delay differential equation, Trans. Amer. Math. Soc. 294, 767-774.
29. L.T. Magalhães (1985), Invariant manifolds for functional differential equations close to ordinary differential equations, Funkcial. Ekvac. 28, 57-82.
30. X.R. Mao (1996), Exponential stability of nonlinear differential delay equations, Systems and Control Letters 28, 159-165.
31. M. Pituk (2003), More on scalar functional differential equations generating a monotone semiflow, Preprint No. 100, Dept. Math. Computing, University of Veszprém, Hungary.
32. M. Pituk (2002), Convergence to equilibria in a differential equation with small delay, Mathematica Bohemica 127, 293-299.
33. M. Pituk (2003), Convergence to equilibria in scalar nonquasimonotone functional differential equations, J. Diff. Eqns 193, 95-130.
34. Y.X. Qin (1958), On the equivalence of stability of differential and differential difference equations, Acta Math. Sinica 8, 47-469. (in Chinese)
35. H.L. Smith (1995), Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. Amer. Math. Soc.
36. H.L. Smith and H.R. Thieme (1990), Monotone semiflows in scalar non-quasi-monotone functional differential equations, Amer. Math. Anal. Applns. 150, 289-306.
37. T. Yoshizawa (1966), Stability Theory by Lyapunov's Second Method. Math. Soc. Japan.
38. Yu.A. Ryabov (1965), Certain asymptotic properties of linear systems with small time lag, Trudy Sem. Teor. Diff. Uravnenii s Otklon. Argumentom Univ. Druzby Narodov Patrisa Lumumby 50, 153-164. (in Russian)
39. B.R. Xu (1997), Stability robustness bounds for linear systems with multiple time-varying delayed perturbations, International J. Systems Sci. 28, 1311-1317.
40. T. Yoneyama (1987), On the 3/2 stability theorem for one-dimensional delay-differential equations, J. Math. Anal. Applns. 125, 161-173.