# American Institute of Mathematical Sciences

July  2005, 13(4): 1007-1034. doi: 10.3934/dcds.2005.13.1007

## Robustness of asymptotic stability to small time delays

 1 Department of Mathematics and Information Science, Yantai University, Yantai 264005 2 FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M.

Received  July 2004 Revised  April 2005 Published  August 2005

The robustness of asymptotic stability properties of ordinary differential equations with respect to small constant time delays is investigated. First, a local robustness result is established for compact asymptotically stable sets of systems with nonlinearities which need be only continuous, so the solutions may even be non-unique. The proof is based on the total stability of the differential inclusion obtained by inflating the original system. Using this first result, it is shown that an exponentially asymptotically stable equilibrium of a nonlinear equation which is Lipschitz in a neighborhood of the equilibrium remains exponentially asymptotically stable under small time delays. Then a global result regarding robustness of exponential dissipativity to small time delays is established with the help of a Lyapunov function for nonlinear systems which satisfy a global Lipschitz condition. The extension of these results to variable time delays is indicated. Finally, conditions ensuring the continuous convergence of the delay system attractors to the attractor of the system without delays are presented.
Citation: Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007
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