OBSERVATION AND CONTROL OF VIBRATIONS IN TREE-SHAPED NETWORKS OF STRINGS ∗ REN ´ ED ´ AGER † Abstract. In this paper we study the controllability problem for a system that models the vibrations of a controlled tree-shaped network of vibrating elastic strings. The control acts through one of the exterior nodes of the network. With the help of the d’Alembert representation formula for the solutions of the 1-d wave equation we find certain linear relations between the traces of the solutions at the nodes of the network. These relations allow to prove a weighted observability inequality with weights that may be explicitly computed in terms of the eigenvalues of the associated elliptic problem. We characterize the class of trees for which all the those weights are different from zero what leads to the spectral controllability of the system. Besides, we consider the same one-node control problem for several networks that are controlled simultaneously. Key words. controllability, observability, string network, wave equation AMS subject classifications. 35L05,35J05,35L20. Introduction. In this paper we study the one-node controllability property of the vibrations of a planar network (i.e., several strings connected at their ends) of elastic homogeneous strings. That is, we analyze the possibility of driving to rest the motion of the network, produced by an initial deformation of its strings, by means of a control applied through one of the nodes. The network considered here coincides at rest with a planar, finite, connected graph without closed paths, whose edges are straight segments. The deformations of the strings of the network are assumed to be transversal to the plane determined by the rest graph. The deformation of every string is expressed by means of a scalar function, defined on the edge of the graph that corresponds to the string and satisfying the 1-d wave equation. At the interior nodes of the network, i.e., those where the strings are coupled, it is assumed that the displacements of all the strings coincide and that the sum of their tensions is equal to zero. These conditions express the continuity of the network and the balance of forces at the junction points. A control acts on one of the exterior nodes that regulates its displacement. The remaining exterior nodes are supposed to be clamped, that is, their displacements are equal to zero. The main results of this paper are related to possibility of proving weighted observability inequal- ities for the solutions of the system modelling the network. Under certain conditions imposed over the lengths of the strings of the networks, which are verified in a generic sense, those observability results allows to obtain information on the controllability properties of the tree-shaped networks from one of exterior node. 1. Notations and statement of the problem. 1.1. Notations for the elements of the graph. In this section, we introduce precise notations for the elements of the rest configuration graph. This is needed to write the equations of the motion of the network in a way that takes into account the topological structure of the graph. Let A be a planar, connected graph without closed paths. According to the usual terminology in Graph Theory, those graphs will be called trees. By the multiplicity of a vertex of A we mean the number of edges that branch out from that vertex. If the multiplicity is equal to one, the vertex is called exterior, otherwise, it is said to be interior. We assume that the graph A does not contain vertices of multiplicity two, since they are irrelevant for our model. * This work has been partially supported by grants PB96-0663 of the DGES (Spain) and the EU project “Homoge- nization and Multiple Scales”. † Departamento de Mat´ ematica Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain (rene dager@mat.ucm.es). 1