A Uniqueness Result for the Linear System of Elasticity and Its Control Theoretical Consequences
Enrique Zuazua · SIAM Journal on Control and Optimization · 1996
We consider the time-dependent linear system of three-dimensional elasticity for an homogeneous and isotropic elastic body $\Omega $ with Dirichlet boundary conditions. We prove that when the domain $\Omega $ is of class $C^2 $ and is not symmetric with respect to a plane parallel to $x_3 = 0$, any solution whose first two components vanish in some open subset of $\Omega $ for a large enough time interval has to be identically zero. The proof of this uniqueness result can be reduced to show that solutions of the scalar wave equation with zero Dirichlet boundary conditions that are of the form $u(x,y,z,t) = \varphi (x,y,t) + \psi (z,t)$ have to be identically zero. This result depends not only on the length of the time interval but also on the geometry of the domain. As a consequence of this uniqueness result we prove that the linear system of elasticity is approximately controllable by means of planar volume forces with zero third component. We show that these results fail for some Lipschitz domains that are piecewise smooth and symmetric with respect to the plane $x_3 = 0$.