This paper deals with a scalar conservation law in 1-D space dimension,
and in particular, the focus is on the stability analysis for such an
equation. The problem of feedback stabilization under
proportional-integral-derivative (PID for short) boundary control is
addressed. In the proportional-integral (PI for short) controller case,
by spectral analysis, the authors provide a complete characterization
of the set of stabilizing feedback parameters, and determine the
corresponding time delay stability interval. Moreover, the stability of
the equilibrium is discussed by Lyapunov function techniques, and by
this approach the exponential stability when a damping term is added to
the classical PI controller scheme is proved. Also, based on Pontryagin
results on stability for quasipolynomials, it is shown that the
closed-loop system subject to PID control is always unstable.