# American Institute of Mathematical Sciences

December  2012, 2(4): 429-455. doi: 10.3934/mcrf.2012.2.429

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

 1 Mathematical Neuroscience Laboratory, CIRB-Collège de France and BANG Laboratory, INRIA Paris-Rocquencourt, 11, place Marcelin Berthelot, 75005 Paris, France

Received  February 2012 Revised  July 2012 Published  October 2012

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.
Citation: Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429
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##### References:
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