This note describes a new framework for the analysis and synthesis of control systems, which constitutes a genuine continuous-time extension of results that are only available in discrete time. In contrast to earlier results the proposed methods involve a specific transformation on the Lyapunov variables and a reciprocal variant of the Projection Lemma, in addition to the classical linearizing transformations on the controller data. For a wide range of problems including robust analysis and synthesis, multi-channel H-2 state- and output-feedback syntheses, the approach leads to potentially less conservative linear matrix inequality (LMI) characterizations. This comes from the fact that the technical restriction of using a single Lyapunov function is to some extent ruled out in this new approach. Moreover, the approach offers new potentials for problems that cannot be handled using earlier techniques. An important instance is the eigenstructure assignment problem blended with Lyapunov-type constraints which is given a simple and tractable formulation.