This paper is concerned with an optimal control problem where the state is constrained to stay either in a smooth open set Omega or in its closure <(Omega)over bar>. Under a "higher-order" sufficient condition for the viability of Omega and <(Omega)over bar>. we prove that the optimal cost function upsilon(Omega) is the unique continuous constrained solution of the Hamilton-Jacobi-Bellman equation. Furthermore, we show that upsilon(Omega) coincides with the optimal cost function upsilon(<(Omega)over) (bar>) on Omega.