  We consider a singular control problem with regime switching that arises in
problems of optimal investment decisions of cash-constrained firms. The value
function is proved to be the unique viscosity solution of the associated
Hamilton-Jacobi-Bellman equation.
  Moreover, we give regularity properties of the value function as well as a
description of the shape of the control regions. Based on these theoretical
results, a numerical deterministic approximation of the related HJB variational
inequality is provided. We finally show that this numerical approximation
converges to the value function. This allows us to describe the investment and
dividend optimal policies.
