Controllability properties of a vibrating string with variable axial load

Abstract

We show that the set of equilibrium-like states (yd, 0) of a vibrating string which can approximately be reached in the energy space H 01(0, 1) × L2(0, 1) from almost any non-zero initial datum by varying its axial load is dense in the subspace H 01(0, 1) × {0} of this space. Our result is based on a constructive argument and makes use of piecewise constant-in-time control functions (loads) only, which enter the model equation as coefficients.