Controllability of tree-shaped networks of vibrating strings

We consider a tree-shaped network of vibrating strings. The problem of cont rollability when the control acts on the root of the tree is analyzed. We g ive a necessary and sufficient condition for the approximate controllabilit y of the system in time 2(l(1) + ... +l(M)) where l(1),...,l(M) denote the lengths of the strings of the network. This is a non-degeneracy condition g uaranteeing that the spectra of any two subtrees of the network connected a t a multiple node are mutually disjoint. (C) 2001 Academie des sciences/Edi tions scientifiques et medicales Elsevier SAS.