This paper deals with an optimal control problem of deterministic two-machine flowshops. Since the sizes of both internal and external buffers are practically finite, the problem is one with state constraints. The Hamilton-Jacobi-Bellman (HJB) equations of the problem involve complicated boundary conditions due to the presence of the state constraints, and as a consequence the usual "verification theorem" may not work for the problem. To overcome this difficulty, it is shown that any function satisfying the HJB equations in the interior of the state constraint domain must be majorized by the value function. The main techniques employed are the "constraint domain approximation" approach and the "weak-Lipschitz" property of the value functions developed in preceding papers. Based on this, an explicit optimal feedback control for the problem is obtained.