Györi, István; Hartung, Ferenc Exponential stability of a state-dependent delay system. (English) Zbl 1144.34051 Discrete Contin. Dyn. Syst. 18, No. 4, 773-791 (2007). This paper deals with the exponential stability of the trivial solution to the state-dependent delayed system \[ x'(t)= \sum_{i=1}^m A_i(t) x(t-\tau _i (t, x_t)). \tag{1} \]It is shown that, under mild assumptions, the trivial solution to (1) is exponentially stable if and only if the trivial solution to the associated linearized time-dependent delayed system\[ y'(t)= \sum_{i=1}^m A_i(t) y(t-\tau _i (t,0)) \]is exponentially stable. The order of both exponential stabilities is compared. As an application, a necessary and sufficient condition for the exponential stability of the trivial solution to a class of threshold-type delayed systems is formulated. Reviewer: Eva Sanchez (Madrid) Cited in 19 Documents MSC: 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations Keywords:state-dependent delay; exponential stability; order of exponential stability; threshold delay PDFBibTeX XMLCite \textit{I. Györi} and \textit{F. Hartung}, Discrete Contin. Dyn. Syst. 18, No. 4, 773--791 (2007; Zbl 1144.34051) Full Text: DOI