Stability properties of black holes in self-gravitating nonlinear electrodynamics

We analyze the dynamical stability of black hole solutions in self-gravitating nonlinear electrodynamics with respect to arbitrary linear fluctuations of the metric and electromagnetic field. In particular, we derive simple conditions on the electromagnetic Lagrangian which imply linear stability in the domain of outer communication. We show that these conditions hold for several of the regular black hole solutions found by Ay\'on-Beato and Garc\'a.