Lyapunov stability of linear predictor feedback for time-varying input delay

For LTI systems with a time-varying input delay, an explicit formula for predictor feedback was presented by Nihtila in 1991. In this note we construct a time-varying Lyapunov functional for the closed-loop system and establish exponential stability. The key challenge is the selection of a state for a transport PDE, which has a non-constant propagation speed, and which is the basis of the stability analysis. We illustrate the design and its conditions with several examples. We also develop an observer equivalent of the predictor feedback design, for the case of time-varying sensor delay.