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November  2007, 18(4): 773-791. doi: 10.3934/dcds.2007.18.773

Exponential stability of a state-dependent delay system

István Györi 1, and  Ferenc Hartung 1,

1. 

Department of Mathematics and Computing, University of Pannonia, H-8201 Veszprém, P.O.Box 158, Hungary, Hungary

Received  May 2006 Revised  February 2007 Published  May 2007

In this paper we study exponential stability of the trivial solution of the state-dependent delay system $\dot x(t)=\sum_{i=1}^m A_{i}(t)x(t-\tau_{i}(t,x_t))$. We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system $\dot y(t)=\sum_{i=1}^m A_{i}(t)y(t-\tau_{i}(t, 0))$ is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.
Keywords: state-dependent delay, threshold delay., exponential stability, order of exponential stability.
Mathematics Subject Classification: Primary: 34K20 Secondary: 34K2.
Citation: István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
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