Controllability for a string with attached masses and Riesz bases for asymmetric spaces

Abstract

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control of Dirichlet, or Neumann, or mixed type is at the left end, and the string is fixed at the right end. Singularities in waves are 'smoothed' out to one order as they cross a point mass. We characterize the reachable set for an $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results. The results are sharp with respect to both the regularity of the solution and with respect to time. The eigenfunctions of the associated Sturm--Liouville problem are used to construct Riesz bases for a family of asymmetric spaces that include the sets of reachable positions and velocities.