  We present a novel penalty approach for a class of quasi-variational
inequalities (QVIs) involving monotone systems and interconnected obstacles. We
show that for any given positive switching cost, the solutions of the penalized
equations converge monotonically to those of the QVIs. We estimate the
penalization errors and are able to deduce that the optimal switching regions
are constructed exactly. We further demonstrate that as the switching cost
tends to zero, the QVI degenerates into an equation of HJB type, which is
approximated by the penalized equation at the same order (up to a log factor)
as that for positive switching cost. Numerical experiments on optimal switching
problems are presented to illustrate the theoretical results and to demonstrate
the effectiveness of the method.
