  Can a black hole that suffers a superradiant instability evolve towards a
'hairy' configuration which is stable? We address this question in the context
of Einstein-charged scalar field theory. First, we describe a family of static
black hole solutions which possess charged scalar-field hair confined within a
mirror-like boundary. Next, we derive a set of equations which govern the
linear, spherically symmetric perturbations of these hairy solutions. We
present numerical evidence which suggests that, unlike the vacuum solutions,
the (single-node) hairy solutions are stable under linear perturbations. Thus,
it is plausible that stable hairy black holes represent the end-point of the
superradiant instability of electrically-charged Reissner-Nordstrom black holes
in a cavity; we outline ways to explore this hypothesis.
