  This paper proposes penalty schemes for a class of weakly coupled systems of
Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from
stochastic hybrid control problems of regime-switching models with both
continuous and impulse controls. We show that the solutions of the penalized
equations converge monotonically to those of the HJBQVIs. We further establish
that the schemes are half-order accurate for HJBQVIs with Lipschitz
coefficients, and first-order accurate for equations with more regular
coefficients. Moreover, we construct the action regions and optimal impulse
controls based on the error estimates and the penalized solutions. The penalty
schemes and convergence results are then extended to HJBQVIs with possibly
negative impulse costs. We also demonstrate the convergence of monotone
discretizations of the penalized equations, and establish that policy iteration
applied to the discrete equation is monotonically convergent with an arbitrary
initial guess in an infinite dimensional setting. Numerical examples for
infinite-horizon optimal switching problems are presented to illustrate the
effectiveness of the penalty schemes over the conventional direct control
scheme.
