Stability analysis for systems with time-varying delay: Trajectory based approach

Recent work by Mazenc and Malisoff provided a trajectory-based approach to proving stability of time-varying systems with time-varying delay. It uses a contraction lemma, instead of Lyapunov-Krasovskii or Razumikhin functions. Here, we use their lemma, and a Lyapunov function for an undelayed system, to provide a new method to prove stability of linear continuous-time, time-varying systems with time-varying bounded delays. No constraint on the upper bound of delay is imposed, nor do we need any differentiability of the delay. Instead, we use an upper bound on an integral average of the delay. We prove input-to-state stability under disturbances.