Controllability properties of a hyperbolic system with dynamic boundary conditions

In this article, we consider a system consisting of two elastic strings with attached tip masses coupled through an elastic spring. Our aim is to analyze its exact boundary controllability properties and to characterize the spaces of controllable initial data depending on the number of controls acting on the boundaries of the strings. We show that singularities in waves are “smoothed by three orders” as they cross a point mass. Consequently, when only one control acts on the extremity of the first string, the space of controlled initial data is asymmetric, the components corresponding to the second string having to be more regular than those corresponding to the first one. Roughly speaking, if the initial data for the string which is directly controlled can be in L^2× H^-1 , they should be at least in H^3× H^2 for the second string, located on the other part of the masses.