We provide closed formulae for the construction of observers and controllers for linear time-delay systems, within the framework of Lyapunov-Krasovskii functionals of complete type. A main feature of the proposed observers and controllers is that they depend on the full state of the system, resulting in a feedback with a distributed term, and they directly emerge from the inverse of a positive definite operator defined on a Hilbert space, determined by the so-called delay Lyapunov matrix. This latter feature brings the proposed approach on par with well-established results for systems without delays, in which Lyapunov matrices serve not only for system analysis but also for control synthesis.