Exact Controllability and Stabilization of a Vibrating String with an Interior Point Mass

In this article we examine the problems of boundary control and stabilization for a one-dimensional wave equation with interior point masses. We show that singularities in waves are “smoothed one order“ as they cross a point mass. Thus in the case of one interior point mass. with, e.g., L²-Dirichlet control at the left end, the most general reachable space (from 0) that one can expect is L² × H- 1 to the left of the mass and H¹ × L² to the right of the mass. We show that this is in fact the optimal result (modulo certain compatibility conditions). Several related results for both control and stabilization of such systems are also given.