Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string

Abstract

We study the large time behavior of the solutions of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Neuman boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term. We give a complete characterization of the positions of the actuator for which the system becomes exponentially stable in the energy space. In the case of nonexponential decay in the energy space we give explicit polynomial decay estimates valid for regular initial data. Moreover we show that the fastest decay rate is obtained if the actuator is located at the middle point of the string.