  This paper is concerned with a new optimization problem named "phase change
rate maximization" for single-input-single-output linear time-invariant
systems. The problem relates to two control problems, namely robust instability
analysis against stable perturbations and minimum-norm strong stabilization. We
define an index of the instability margin called "robust instability radius
(RIR)" as the smallest $H_\infty$-norm of a stable perturbation that stabilizes
a given unstable system. This paper has two main contributions. It is first
shown that the problem of finding the exact RIR via the small-gain condition
can be transformed into the problem of maximizing the phase change rate at the
peak frequency with a phase constraint. Then, we show that the maximum is
attained by a constant or a first-order all-pass function and derive
conditions, under which the RIR can be exactly characterized, in terms of the
phase change rate. Two practical applications are provided to illustrate the
utility of our results.
