Boundary controllability of linear elastodynamic systems

We consider the problem of controlling the motion of a linear elastodynamic system by the application of traction forces on a small subset of the boundary surface. We show that the system is boundary controllable in time T in the sense that the set of all states, reachable at time T from the zero state, is dense in the Hilbert space of finite energy states if T is greater than twice a certain time T₂. We also show that the system is not boundary controllable in time T if T < 2T₁ where T₁ < T₂. The boundary controllability problem is shown to be the dual of the boundary observability problem where we attempt to determine an initial state of an elastic system by observing the velocities on a subset of the boundary over a time interval (0, T).