Some new applications of Russell's principle to infinite dimensional vibrating systems

The aim of this work is to highlight the interest of a by now classical methodology, commonly called Russell's principle, in proposing new control strategies and estimates for infinite dimensional vibrating systems. After describing (with complete proofs) a particular version, of interest for our work, of Russell's principle, we consider two main applications. The first one, which mainly contains results which are new, studies the approximation of a class of boundary control systems by systems with controls distributed in an open set (internal controls), with the support shrinking to the boundary. These approximations are interesting since for the approximating systems we have bounded input operators, which makes easier the use of many control theoretic tools. The second application concerns the approximation of exact controls for infinite dimensional systems using their projections on finite dimensional spaces. We propose here an alternative, based on Russell's principle, of the existing approximation methods, often based on inverting the Gramian.