Feedback Stabilization for a Scalar Conservation Law with PID Boundary Control

This paper deals with a scalar conservation law in 1-D spacedimension, and in particular, the focus is on the stability analysisfor such an equation. The problem of feedback stabilization underproportional-integral-derivative (PID for short) boundary control isaddressed. In the proportional-integral (PI for short) controllercase, by spectral analysis, the authors provide a completecharacterization of the set of stabilizing feedback parameters, anddetermine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov functiontechniques, and by this approach the exponential stability when adamping term is added to the classical PI controller scheme isproved. Also, based on Pontryagin results on stability forquasipolynomials, it is shown that the closed-loop system subject toPID control is always unstable.