The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity   Solution Approach
  In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market. 
Brahim El Asri and Said Hamadene