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

Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain