We study a discounted infinite horizon control problem in a stratified setting. We write down an appropriate Hamilton-Jacobi-Bellman equation and prove a comparison principle implying that the value function of the system is the unique bounded continuous viscosity solution. We specifycally consider the state variable space partitioned in two open sets plus a common boundary, dubbed interface, assumed to be a C2 hypersurface. Further, we provide each open region of a control system with dynamics, cost and discount factor. To glue them on the interface, and build the integrated system we are interested on, we follow ideas introduced in a recent paper of Barles and coauthors. The crucial point is to determine the appropriate Hamilton-Jacobi-Bellman equation on the interface, notice that Ishii theory does not seem suitable for our model. We just consider, at least for subsolutions, a tangential equation, whose Hamiltonian contains controls related to directions admissible with respect to the interface in a suitable sense. Our techniques are of dynamical type, we associate to any trajectory of the integrated system an index, apparently not previously recorded in the literature, which provides in a sense information about their behavior around the interface. We construct on it a couple of arguments by induction, crucial to establish our main results.