Approximate controllability of coupled 1-d wave equations onstar-shaped graphs

Abstract

In this Note, we study the approximate controllability of a cascade system of two 1-d wave equations defined on a star-shaped planar graph. Only the first equation is controlled, with controls applied at the simple vertices. The controls act on the second equation through the coupling at the multiple vertex. We give a necessary and sufficient condition for the approximate controllability of the system: all the ratios of the lengths of any two different edges of the graph should be irrational numbers. The result implies that, under these conditions, the approximate desensitizing controllability of a star-shaped network of strings holds, when the desensitized functional is the L2-norm of the displacement of the multiple node.