This is the first of two papers regarding a family of linear convex control problems in Hilbert spaces and the related Hamilton-Jacobi-Bellman equations. The framework is motivated by an application to boundary control of a PDE modeling investments with vintage capital. Existence and uniqueness of a strong solution (namely, the limit of classic solutions of approximating equations, introduced by Barbu and Da Prato) is investigated. Moreover, such a solution is proved to be cl in the space variable.