  We investigate the well-posedness of the characteristic initial-boundary
value problem for the Einstein equations in Bondi-like coordinates (including
Bondi, double-null and affine). We propose a definition of strong hyperbolicity
of a system of partial differential equations of any order, and show that the
Einstein equations in Bondi-like coordinates in their second-order form used in
numerical relativity do not meet it, in agreement with results of
Giannakopoulos et al for specific first-order reductions. In the principal
part, frozen coefficient approximation that one uses to examine hyperbolicity,
we explicitly construct the general solution to identify the solutions that
obstruct strong hyperbolicity. Independently, we present a first-order
symmetric hyperbolic formulation of the Einstein equations in Bondi gauge,
linearised about Schwarzschild, thus completing work by Frittelli. This
establishes an energy norm ($L^2$ in the metric perturbations and selected
first and second derivatives), in which the initial-boundary value problem,
with initial data on an outgoing null cone and boundary data on a timelike
cylinder or an ingoing null cone, is well-posed, thus verifying a conjecture by
Giannakopoulos et al. Unfortunately, our method does not extend to the pure
initial-value problem on a null cone with regular vertex.
