On the Dynamics of Nonautonomous General Dynamical Systems and Differential Inclusions
Tóm tắt
This paper is concerned with the dynamics of nonautonomous general dynamical systems (NAGDSs in short) and applications to differential inclusions on ℝ
m
. First, we show that if a NAGDS has a compact uniformly attracting set, then it has a pullback attractor
$\mathcal{A}$
with the parametrically inflated pullback attractor
$\mathcal{A}(\varepsilon_0)$
being uniformly forward attracting. Then, we establish some stability results for pullback attractors. Finally, we apply the abstract theory to nonautonomous differential inclusions on ℝ
m
to obtain some interesting results. In particular, the effects of small time delays to asymptotic stability is addressed.
Tài liệu tham khảo
Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)
Caraballo, T., Langa, J.A.: Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise. J. Math. Anal. Appl. 260, 602–622 (2001)
Caraballo, T., Marin-Rubio, P., Robinson, J.C.: A comparison between two theories for multivalued semiflows and their asymptotic behavior. Set-Valued Anal. 11, 297–322 (2003)
Caraballo, T., Marin-Rubio, P., Valero, J.: Autonomous and nonautonomous attractors for differential equations with delays. J. Differential Equations 208, 9–41 (2005)
Cheban, D.N., Mammana, C.: Relation between different types of global attractors of set-valued nonautonomous dynamical systems. Set-Valued Anal. 13, 291–321 (2002)
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J.: Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149, 69–114 (1998)
Colonius, F., Kliemann, W.: Limits of input-to-state stability. Systems Control Lett. 49, 111–120 (2003)
Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin (1998)
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Nauka, Moscow (1985) (in Russian)
Kharitonov, V.L.: Robust stability analysis of time delay systems. A Survey Ann. Rev. Control 23, 185–196 (1999)
Kloeden, P.E.: Asymptotic invariance and limit sets of general control systems. J. Differential Equations 19, 91–105 (1975)
Kloeden, P.E., Schmalfuss, B.: Asymptotic behaviour of nonautonomous difference inclusions. Systems Control Lett. 33, 275–280 (1998)
Li, D.S.: On dynamical stability in general dynamical systems. J. Math. Anal. Appl. 263, 455–478 (2001)
Li, D.S.: Morse decompositions for general dynamical systems and differential inclusions with applications to control systems. SIAM J. Control Optim. 46, 35–60 (2007)
Li, D.S., Zhang, X.X.: On the stability in GDSs and differential inclusions. J. Math. Anal. Appl. 274, 705–724 (2002)
Li, D.S., Kloeden, P.E.: Robustness of asymptotic stability to small time delays. Discrete Contin. Dynam. Systems 13, 1007–1034 (2005)
Li, D.S., Kloeden, P.E.: On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions. J. Differential Equations 224, 1–38 (2006)
Logemann, H., Rebarber, R., Weiss, G.: 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 (1996)
Li, T.: Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Commun. Pure Appl. Anal. 6, 279–285 (2007)
Pituk, M.: Convergence to equilibria in scalar nonquasimonotone functional differential equations. J. Differential Equations 193, 95–130 (2003)
Wang, Y.J., Li, D.S., Kloeden, P.E.: On the asymptotical behavior of nonautonomous dynamical systems. Nonlinear Anal. TMA 59, 35–53 (2004)
Wang, Y.J., Zhou, S.F.: Kernel sections and uniform attractors of multi-valued semiprocesses. J. Differential Equations 232, 573–622 (2007)