  In this paper, we initiate the study of the asymptotically AdS
initial-boundary value problem for the Einstein-massless Vlasov system with
$\Lambda<0$ in spherical symmetry. We will establish the existence and
uniqueness of a maximal future development for the characteristic
initial-boundary value problem in the case when smooth initial data are
prescribed on a future light cone $\mathcal{C}^{+}$ emanating from a point at
$\{r=0\}$ and a reflecting boundary condition is imposed on conformal infinity
$\mathcal{I}$. We will then prove a number of continuation criteria for smooth
solutions of the spherically symmetric Einstein-massless Vlasov system, under
the condition that the ratio $2m/r$ remains small in a neighborhood of
$\{r=0\}$. Finally, we will establish a Cauchy stability statement for Anti-de
Sitter spacetime as a solution of the spherically symmetric Einstein-massless
Vlasov system under initial perturbations which are small only with respect to
a low regularity, scale invariant norm $||\cdot||$. This result will imply, in
particular, a long time of existence statement for $||\cdot||$-small initial
data.
  This paper provides the necessary tools for addressing the AdS instability
conjecture in the setting of the spherically symmetric Einstein--massless
Vlasov system, a task which is carried out in our companion paper. However, the
results of this paper are also of independent interest.
