Exponential stability of a state-dependent delay system

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.