  Deterministic optimal impulse control problem with terminal state constraint
is considered. Due to the appearance of the terminal state constraint, the
value function might be discontinuous in general. The main contribution of this
paper is the introduction of an intrinsic condition under which the value
function is continuous. Then by a Bellman dynamic programming method, the
corresponding Hamilton-Jacobi-Bellman type quasi-variational inequality (QVI,
for short) is derived for which the value function is a viscosity solution. The
issue of whether the value function is characterized as the unique viscosity
solution to this QVI is carefully addressed and the answer is left open
challengingly.
