  In this article we study a controllability problem for a parabolic and a
hyperbolic partial differential equations in which the control is the shape of
the domain where the equation holds. The quantity to be controlled is the trace
of the solution into an open subdomain and at a given time, when the right hand
side source term is known. The mapping that associates this trace to the shape
of the domain is nonlinear. We show (i) an approximate controllability property
for the linearized parabolic problem and (ii) an exact local controllability
property for the linearized and the nonlinear equations in the hyperbolic case.
We then address the same questions in the context of a finite difference
spatial semi-discretization in both the parabolic and hyperbolic problems. In
this discretized case again we prove a local controllability result for the
parabolic problem, and an exact controllability for the hyperbolic case,
applying a local surjectivity theorem together with a unique continuation
property of the underlying adjoint discrete system.
