Exact controllability for string with attached masses

TL;DR

This paper investigates the exact boundary controllability of a vibrating string with multiple interior point masses, characterizing the reachable states and solving the control problem via a moment problem approach.

Contribution

It introduces a novel analysis of wave singularities at point masses and reduces the control problem to a solvable moment problem using exponential divided differences.

Findings

Reachable set characterized for $L^2$ control

Control problem reduced to a moment problem

Unique shape and velocity controllability established

Abstract

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control 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 a $L^2$ 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.