Controllability of star-shaped networks of strings

In this note we consider a star-shaped network of vibrating strings. The pr oblem of controllability when one control acts on the junction point is con sidered. A simple proof is given that, in particular, does not use Ingham i nequalities, of the fact that the set of reachable data is dense, whenever the lengths of the strings are mutually irrational. The proof is based on a n observability inequality with suitable weights on the Fourier coefficient s that is easily obtained as a consequence of the time-periodicity of the s olutions. Those weights can be estimated under certain algebraicity conditi ons imposed on the lengths of the strings. This allows to proof exact contr ollability results in Sobolev spaces of appropriate order. Further results are also presented concerning the control from a free end and diffusion pro cesses. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.