Controllability Properties of a Vibrating String with Variable Axial Load Only

Abstract

We show that the set of equilibrium-like states (y_d,0) of a vibrating string which can approximately be reached in the energy space H¹ ₀ (0,1)× L²(0,1) from almost any non-zero initial datum, namely, (y₀,y₁) ε (H²(0,1)∩H¹ ₀(0,1))× H¹(0,1), (y₀,y₁) ≠ (0,0) by varying its axial load only is dense in the subspace H¹ ₀ (0,1)× {0} of this space. This 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.