# 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 and Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429
##### References:
 [1] G. Allaire, "Conception Optimale de Structures,'' Springer-Verlag, New York, 2007. [2] Haim Brezis, "Analyse Fonctionnelle,'' Masson, Paris, 1983. [3] C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method, Numerische Mathematik, 102 (2006), 413-462. doi: 10.1007/s00211-005-0651-0. [4] J. Céa, Numerical methods of optimum shape design, in "Optimization of Distributed Parameter Structure'' (editors, E. Haug and Jean Céa), Alphen aan den Rijn, Sijthoff and Noordhoff, the Netherlands, 1981. [5] J. Céa, Optimization of distributed parameter structures, NATO Advanced study Institutes series, 1981. [6] D. Chenais and E. Zuazua, Controlability of an elliptic equation and its finite difference approximation by the shape of the domain, Numerische Mathematik, 95 (2003), 63-99. doi: 10.1007/s00211-002-0443-8. [7] D. Chenais, On the existence of a solution in a domain identification problem, Journal of Math. Analysis and Applications, 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. [8] E. Holmgren., Über systeme von linearen partiellen differentialgleichungen, Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103. [9] L. Hörmander, "Linear Partial Differential Operators,'' Springer-Verlag, New York, 1969. [10] J. A. Infante and E. Zuazua., Boundary observability for the space semi-discretizations of the 1 - d wave equation, Mathematical Modeling and Numerical Analysis, 33 (1999), 407-438. doi: 10.1051/m2an:1999123. [11] J. L. Lions., "Contrôle Optimal de Systemes Gouvernés par des Équations aux Dérivées Partielles,'' Dunod, Paris, 1968. [12] J. L. Lions., "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués (I and II),'' Masson, Paris, 1988. [13] J. L. Lions, Exact controllability, stabilization and perturbations for distributed sytems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [14] D. G. Luenberger, "Optimization by Vector Space Methods,'' John Wiley and Sons Inc., New York, 1969. [15] F. Murat and J. Simon., Sur le Contrôle par un Domaine Géométrique, Publication du Laboratoire d'analyse numérique, 189, Paris VI, 1976. [16] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-d wave equation, Systems and Control Letters, 48 (2003), 261-280. doi: 10.1016/S0167-6911(02)00271-2. [17] B. Rousselet, Optimization of distributed parameter structures, NATO Advanced Study Institutes Series, 50 (1981), 1474-1501. [18] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Func. Anal. Optim., 2 (1980), 649-687. [19] J. Simon., "Diferenciación con Respecto al Dominio," Lecture notes, Universidad de Sevilla, 1989. [20] J. P. Zolesio., Optimization of distributed parameter structures, NATO Advanced Studies Series, 50 (1981), 1089-1151. [21] E. Zuazua, Some problems and results on the controllability of partial differential equations, in "European Congress of Mathematics," Progress in Mathematics. Birkhäuser, 1996. [22] E. Zuazua, Some new results related to the null controllability of the 1-d heat equation, Seminaire X-EDP, Ecole Polytechnique, VIII (1997-1998), 1-22. [23] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563. [24] E. Zuazua, Observability of the 1-d waves in heterogenous and semi discrete media, In "Advances in Structural Control," CIMNE, 1999. [25] E. Zuazua, Controllability of partial differential equation and its semi-discrete approximations, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.

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##### References:
 [1] G. Allaire, "Conception Optimale de Structures,'' Springer-Verlag, New York, 2007. [2] Haim Brezis, "Analyse Fonctionnelle,'' Masson, Paris, 1983. [3] C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method, Numerische Mathematik, 102 (2006), 413-462. doi: 10.1007/s00211-005-0651-0. [4] J. Céa, Numerical methods of optimum shape design, in "Optimization of Distributed Parameter Structure'' (editors, E. Haug and Jean Céa), Alphen aan den Rijn, Sijthoff and Noordhoff, the Netherlands, 1981. [5] J. Céa, Optimization of distributed parameter structures, NATO Advanced study Institutes series, 1981. [6] D. Chenais and E. Zuazua, Controlability of an elliptic equation and its finite difference approximation by the shape of the domain, Numerische Mathematik, 95 (2003), 63-99. doi: 10.1007/s00211-002-0443-8. [7] D. Chenais, On the existence of a solution in a domain identification problem, Journal of Math. Analysis and Applications, 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. [8] E. Holmgren., Über systeme von linearen partiellen differentialgleichungen, Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103. [9] L. Hörmander, "Linear Partial Differential Operators,'' Springer-Verlag, New York, 1969. [10] J. A. Infante and E. Zuazua., Boundary observability for the space semi-discretizations of the 1 - d wave equation, Mathematical Modeling and Numerical Analysis, 33 (1999), 407-438. doi: 10.1051/m2an:1999123. [11] J. L. Lions., "Contrôle Optimal de Systemes Gouvernés par des Équations aux Dérivées Partielles,'' Dunod, Paris, 1968. [12] J. L. Lions., "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués (I and II),'' Masson, Paris, 1988. [13] J. L. Lions, Exact controllability, stabilization and perturbations for distributed sytems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [14] D. G. Luenberger, "Optimization by Vector Space Methods,'' John Wiley and Sons Inc., New York, 1969. [15] F. Murat and J. Simon., Sur le Contrôle par un Domaine Géométrique, Publication du Laboratoire d'analyse numérique, 189, Paris VI, 1976. [16] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-d wave equation, Systems and Control Letters, 48 (2003), 261-280. doi: 10.1016/S0167-6911(02)00271-2. [17] B. Rousselet, Optimization of distributed parameter structures, NATO Advanced Study Institutes Series, 50 (1981), 1474-1501. [18] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Func. Anal. Optim., 2 (1980), 649-687. [19] J. Simon., "Diferenciación con Respecto al Dominio," Lecture notes, Universidad de Sevilla, 1989. [20] J. P. Zolesio., Optimization of distributed parameter structures, NATO Advanced Studies Series, 50 (1981), 1089-1151. [21] E. Zuazua, Some problems and results on the controllability of partial differential equations, in "European Congress of Mathematics," Progress in Mathematics. Birkhäuser, 1996. [22] E. Zuazua, Some new results related to the null controllability of the 1-d heat equation, Seminaire X-EDP, Ecole Polytechnique, VIII (1997-1998), 1-22. [23] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563. [24] E. Zuazua, Observability of the 1-d waves in heterogenous and semi discrete media, In "Advances in Structural Control," CIMNE, 1999. [25] E. Zuazua, Controllability of partial differential equation and its semi-discrete approximations, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.
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