  We study an optimal switching problem with a state constraint: the controller
is only allowed to choose strategies that keep the controlled diffusion in a
closed domain. We prove that the value function associated with this problem is
the limit of value functions associated with unconstrained switching problems
with penalized coefficients, as the penalization parameter goes to infinity.
This convergence allows to set a dynamic programming principle for the
constrained switching problem. We then prove that the value function is a
solution to a system of variational inequalities (SVI for short) in the
constrained viscosity sense. We finally prove that uniqueness for our SVI
cannot hold and we give a weaker characterization of the value function as the
maximal solution to this SVI. All our results are obtained without any
regularity assumption on the constraint domain.
