The analysis of a class of infinite-dimensional Hamilton-Jacobi-Bellman (HJB) equations is undertaken related to linear convex boundary control problems for PDEs with constraints on the state. A definition of generalized solution (namely, weak solution) of the HJB equation is provided (weak is the limit of strong, while strong is the limit of classical, as defined in [S. Faggian, Appl. Math. Optim., 51 (2005), pp. 123-162]). Consequently an existence and uniqueness result is provided for weak solutions of the HJB equation, as well as existence of an optimal control, being the limit of optimal controls of approximating problems. The study then describes an economic application to optimal investment with vintage capital, having positivity constraints on the capital.