Hamilton–Jacobi–Bellman equations for optimal control processes with convex state constraints

This work aims at studying some optimal control problems with convex state constraint sets. It is known that for state constrained problems, and when the state constraint set coincides with the closure of its interior, the value function satisfies a Hamilton–Jacobi equation in the constrained viscosity sense. This notion of solution has been introduced by H.M. Soner (1986) and provides a characterization of the value functions in many situations where an inward pointing condition (IPC) is satisfied. Here, we first identify a class of control problems where the constrained viscosity notion is still suitable to characterize the value function without requiring the IPC. Moreover, we generalize the notion of constrained viscosity solutions to some situations where the state constraint set has an empty interior. •We provide a characterization of the Value Function of an Optimal Control problem with state constraint sets.•A NFT theorem is proved without requiring any of the so-called Inward/Outward Pointing Conditions.•The technique relies on the convexity of the state constraint set and the graph of the dynamics.•We generalize the notion of constrained viscosity solutions to some situations where the state constraint set has an empty interior.