Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass

Abstract

We consider a hybrid, one-dimensional, linear system consisting in two flexible strings connected by a point mass. It is known that this system presents two interesting features. First, it is wellposed in an asymmetric space in which solutions have one more degree of regularity to one side of the point mass. Second, that the spectral gap vanishes asymptotically. We prove that the first property is a consequence of the second one. We also consider a system in which the point mass is replaced by a string of length 2ε and density 1/2ε. We show that, as ε → 0, thesolutions of this system converge to those of the original one. We also analyze the convergence of the spectrum and obtain the well-posedness of the limit system in the asymmetric space as a consequence of non-standard uniform bounds of solutions of the approximate problems. Finally we consider the controllability problem. It is well known that the limit system with L-controls on one end is exactly controllable in an asymmetric space. We show how this result can be obtained as the limit when ε → 0 of partial controllability results for the approximate systems inwhich the number of controlled frequencies converges to infinity as ε → 0. This is done by means of some new results on non-harmonic Fourier series.