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Exponential stability of a state-dependent delay system. (English) Zbl 1144.34051

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.

MSC:

34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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