  We consider Einstein-Maxwell-dilaton gravity with the non-minimal exponential
coupling between the dilaton and the Maxwell field emerging from low energy
heterotic string theory. The dilaton is endowed with a potential that
originates from an electromagnetic Fayet-Iliopoulos (FI) term in
$\mathcal{N}=2$ extended supergravity in four spacetime dimensions. For the
case we are interested in, this potential introduces a single parameter
$\alpha$. When $\alpha\rightarrow 0$, the static black holes (BHs) of the model
are the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solutions. When
$\alpha \rightarrow \infty$, the BHs become the standard Reissner-Nordstr\"om
(RN) solutions of electrovacuum General Relativity. The BH solutions for finite
non-zero $\alpha$ interpolate between these two families. In this case, the
dilaton potential regularizes the extremal limit of the GMGHS solution yielding
a set of zero temperature BHs with a near horizon $AdS_2\times S^2$ geometry.
We show that, in the neighborhood of these extremal solutions, there is a
subset of BHs that are dynamically and thermodynamically stable, all of which
have charge to mass ratio larger than unity. By dynamical stability we mean
that no growing quasi-normal modes are found; thus they are stable against
linear perturbations (spherical and non-spherical). Moreover, non-linear
numerical evolutions lend support to their non-linear stability. By
thermodynamical stability we mean the BHs are stable both in the canonical and
grand-canonical ensemble. In particular, both the specific heat at constant
charge and the isothermal permittivity are positive. This is not possible for
RN and GMGHS BHs. We discuss the different thermodynamical phases for the BHs
in this model and comment on what may allow the existence of both dynamically
and thermodynamically stable BHs.
