Robustness of asymptotic stability to small time delays
Robustness of asymptotic stability to small time delays
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DOI：
10.3934/dcds.2005.13.1007
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发表时间：
2005-08
影响因子：
1.1
通讯作者：
Desheng Li;P. Kloeden
中科院分区：
文献类型：
--
作者：
Desheng Li;P. Kloeden
研究了常微分方程渐近稳定性关于小时滞的稳健性。首先，对于仅需要连续的非线性系统的紧渐近稳定集，建立了一个局部鲁棒性结果，因此解甚至可能是非唯一的。证明是基于通过膨胀原系统得到的微分包含的总稳定性。利用第一个结果，证明了一类非线性方程在平衡点邻域内的Lipschitz指数渐近稳定平衡点在小时滞下保持指数渐近稳定。然后，对于满足全局Lipschitz条件的非线性系统，借助于Lyapunov函数，建立了指数耗散性对小时滞的鲁棒性的全局结果。指出了这些结果对变时滞的推广。最后，给出了时滞系统吸引子连续收敛于无时滞系统吸引子的条件。
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.