Shape Control by Collinear Actuators

Abstract

We consider a two-dimensional elastic medium bounded by a free surface. The medium deforms under the influence of actuators, which are modelled as supplying a unidirectional stress. The goal of the control mechanism provided by the actuators is to deform the body into a given shape. In a linearized situation, this means achieving a given normal displacement on the boundary. We prove that, for generic domains with smooth boundary, every normal displacement of the boundary can be achieved. We also consider “stress-free” states, for which the sum of the elastic stresses in the medium and the stress provided by the actuators vanishes. The study of these states leads to a hyperbolic boundary value problem.