The goal in this paper is to explore the connections between the boundary condition formulation and the reflected Hamiltonian formulation more directly, without recourse to uniqueness results for solutions or a restriction to solutions which can be interpreted as value functions for some control problem. The equivalence of the two notions of solution is considered. It is quite simple to show that a solution in the reflected Hamiltonian sense must be a solution in the boundary condition sense; the two notions of subsolutions are in fact equivalent.
To prove the analogous result for supersolutions, the author needs to make stronger affine-convex assumptions on the dynamics and running cost. These are satisfied in the recent applications in queueing theory. At points of differentiability, equivalent conditions for the boundary conditions are given in terms of the Hamiltonian and the geometry of the state trajectories using optimal controls.