小时滞梯度系统的动力学行为

Abstract

Abstract:

In this article, we investigate the dynamical behavior of the following general nonlinear gradient-like evolutionary equation with small time delay ∂_tu+Au=f(u(t),u(t-τ)). We prove that each bounded solution of the delayed equation will converge to some equilibrium as t→∞ provided the delay is sufficiently small. This indicates that gradient system with small time delay behaves very much like the nondelayed one. The approach here is mainly based on the Morse structure of invariant sets of gradient system and some geometric analysis of evolutionary equations. The proof of this result is completed in two steps. First, with the hypothesis of gradient system, finite and isolated equilibria, we prove that there exists a sufficiently small delay such that any bounded solution of the delayed equation will ultimately enter and stay in the neighborhood of one equilibrium. Second, with the hypothesis of hyperbolic equilibrium, we utilize exponential dichotomies and a series estimates to prove that there exists ε > 0 and τ > 0 sufficiently small such that any solution of the delayed equation lying in the ε-neighborhood of one equilibrium will converge to this equilibrium as t → ∞.

Key words: Isolated hyperbolic equilibrium, Aubin-Lions lemma, Gradient system, Morse structure, Exponential dichotomies

CLC Number:

O193.4

Yin Xunwu, Li Desheng. Dynamical Behavior of Gradient System with Small Time Delay[J].Acta mathematica scientia,Series A, 2015, 35(3): 464-477.

Trendmd

References

[1] Gu K, Kharitonov V L, Chen J. Stability of Time-Delay Systems. Boston: Birkhäuser, 2003
[2] Han Q L. On stability of linear neutral systems with mixed time-delays: a discretized Lyapunov functional approach. Automatica, 2005, 41: 1209-1218
[3] Guo F M, Yu W, Huang F L. Robustness with respect to small delays for exponential stability of non-autonomous systems. J Math Anal Appl, 2003, 288: 671-679
[4] Arino O, Pituk M. Morse on linear differential systems with small delays. J Diff Eqns, 2005, 170: 381-407
[5] Driver R D. Linear differential systems with small delays. J Diff Eqns, 1976, 21: 149-167
[6] Hale J K, Verduyn S M. Effects of small delays on stability and control. Oper Theory Adv, 2001, 122: 275-301
[7] Xu B R. Stability robustness bounds for linear systems with multiple time-varying delayed perturbations. International J Systems Sci, 1997, 28: 1311-1317
[8] Datko R. Time delay perturbations and robust stability//Elworthy, Everitt. Lecture Notes in Pure and Appl Math, New York: Dekker, 1994: 457-468
[9] Li D, Wang Y. Asymptotic behavior of gradient systems with small time delays. Nonlinear Anal Real World Appl, 2010, 11: 1627-1633
[10] Li D. Morse decompositions for general dynamical systems and differential inclusions with applications to control systems. SIAM J Control Opt, 2007, 46: 35-60
[11] Li D, Kloeden P E. Robustness of asymptotic stability to small time delays. Disc Cont Dyn Syst, 2005, 13: 1007-1034
[12] Friesecke G. Convergence to equilibrium for delay-diffusion equations with small delay. J Dyn Diff Eqns, 1993, 5: 89-103
[13] Kharitonov V L. Robust stability analysis of time delay systems: A survey. Ann Rev Control, 1999, 23: 185-196
[14] Mao X R. Exponential stability of nonlinear differential delay equations. Systems Cont Lett, 1996, 28: 159-165
[15] Pituk M. Convergence to equilibria in a differential equation with small delay. Math Bohem, 2002, 127: 293-299
[16] Pituk M. Convergence to equilibria in scalar nonquasimonotone functional differential equations. J Diff Eqns, 2003, 193: 95-130
[17] Smith H L, Thieme H R. Monotone semiflows in scalar non-quasi-monotone functional differential equations. Amer Math Anal Appl, 1990, 150: 289-306
[18] Henry D. Geometric Theory of Semilinear Parabolic Equations, Vol. 840 of Lecture Notes in Mathematics. Berlin: Springer, 1981
[19] Sell George R, You Yuncheng. Dynamics of Evolutionary Equations. New York: Springer-Verlag, 2002
[20] Deimling K. Nonlinear Functional Analysis. Berlin: Springer, 1985
[21] Showalter R E. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Providence, RI: American Mathematical Society, 1997
[22] Hale J K. Asymptotic Behavior of Dissipative Systems, Vol. 25 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 1988

Dynamical Behavior of Gradient System with Small Time Delay

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 0

Metrics

Comments

Recommended 0