  We introduce the notion of adaptive synchronisation for pushdown automata, in
which there is an external observer who has no knowledge about the current
state of the pushdown automaton, but can observe the contents of the stack. The
observer would then like to decide if it is possible to bring the automaton
from any state into some predetermined state by giving inputs to it in an
\emph{adaptive} manner, i.e., the next input letter to be given can depend on
how the contents of the stack changed after the current input letter. We show
that for non-deterministic pushdown automata, this problem is
2-EXPTIME-complete and for deterministic pushdown automata, we show
EXPTIME-completeness.
  To prove the lower bounds, we first introduce (different variants of)
subset-synchronisation and show that these problems are polynomial-time
equivalent with the adaptive synchronisation problem. We then prove hardness
results for the subset-synchronisation problems. For proving the upper bounds,
we consider the problem of deciding if a given alternating pushdown system has
an accepting run with at most $k$ leaves and we provide an $n^{O(k^2)}$ time
algorithm for this problem.
