Controllability of star-shaped networks of strings

The problem of controlling a network of n vibrating homogeneous strings coupled at a common point in a star-shaped configuration is studied. The contol acts on the network through the extreme of one of the strings. A weighted observability-type inequality is proved. This implies the approximate controllability of the system if the lengths of the uncontrolled strings are mutually rationally independent and the control time is at least twice the sum of the lengths of all the strings. These two conditions are shown to be also necessary for the approximate controllability. Besides, resorting to diophantine approximation theory, some sufficient conditions on the lengths of the strings are given, that allow to identify a space of controllable data that turns out to be constituted by functions that are 'n - 2 + epsilon derivatives' more smooth than those of the usual space of controllable data for the wave equation.