This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.
Keywords: one-dimensional wave equation, null controllability, finite element methods, Carleman estimates
@article{COCV_2013__19_4_1076_0,
author = {C{\^\i}ndea, Nicolae and Fern\'andez-Cara, Enrique and M\"unch, Arnaud},
title = {Numerical controllability of the wave equation through primal methods and {Carleman} estimates},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1076--1108},
year = {2013},
publisher = {EDP Sciences},
volume = {19},
number = {4},
doi = {10.1051/cocv/2013046},
mrnumber = {3182682},
zbl = {1292.35162},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2013046/}
}
TY - JOUR AU - Cîndea, Nicolae AU - Fernández-Cara, Enrique AU - Münch, Arnaud TI - Numerical controllability of the wave equation through primal methods and Carleman estimates JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1076 EP - 1108 VL - 19 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2013046/ DO - 10.1051/cocv/2013046 LA - en ID - COCV_2013__19_4_1076_0 ER -
%0 Journal Article %A Cîndea, Nicolae %A Fernández-Cara, Enrique %A Münch, Arnaud %T Numerical controllability of the wave equation through primal methods and Carleman estimates %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1076-1108 %V 19 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2013046/ %R 10.1051/cocv/2013046 %G en %F COCV_2013__19_4_1076_0
Cîndea, Nicolae; Fernández-Cara, Enrique; Münch, Arnaud. Numerical controllability of the wave equation through primal methods and Carleman estimates. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1076-1108. doi: 10.1051/cocv/2013046
[1] and , An implicit scheme uniformly controllable for the 2-D wave equation on the unit square. J. Optimiz. Theory Appl. 143 (2009) 417-438. | MR
[2] , and , Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR
[3] , Lipschitz stability in an inverse problem for the wave equation, Master report (2001) available at: http://hal.archives-ouvertes.fr/hal-00598876/en/.
[4] , and , Global Carleman estimates for wave and applications. Preprint.
[5] and , Convergence of an inverse problem for discrete wave equations. Preprint.
[6] , and , Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28 (2008) 186-214. | MR
[7] , and , An approximation method for the exact controls of vibrating systems. SIAM. J. Control. Optim. 49 (2011) 1283-1305. | MR
[8] and , Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. SIAM. J. Control. Optim. 48 (2009) 521-550. | MR
[9] and , Experimental study of the HUM control operator for linear waves. Experiment. Math. 19 (2010) 93-120. | MR
[10] and , Convex analysis and variational problems, Classics in Applied Mathematics. Soc. Industr. Appl. Math. SIAM, Philadelphia 28 (1999). | MR
[11] and , The wave equation: Control and numerics. In Control of partial differential equations of Lect. Notes Math. Edited by P.M. Cannarsa and J.M. Coron. CIME Subseries, Springer Verlag (2011). | MR
[12] and , Strong convergent approximations of null controls for the heat equation. Séma Journal 61 (2013) 49-78.
[13] and , Numerical null controllability of the 1-d heat equation: Carleman weights and duality. Preprint (2010). Available at http://hal.archives-ouvertes.fr/hal-00687887.
[14] and , Numerical null controllability of a semi-linear 1D heat via a least squares reformulation. C.R. Acad. Sci. Série I 349 (2011) 867-871.
[15] and , Numerical null controllability of semi-linear 1D heat equations: fixed points, least squares and Newton methods. Math. Control Related Fields 2 (2012) 217-246. | MR
[16] A.V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, in vol. 34 of Lecture Notes Series. Seoul National University, Korea (1996) 1-163. | MR
[17] , , and , Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46 (2007) 1578-1614. | MR
[18] , On Carleman estimates for hyperbolic equations. Asymptotic Analysis 32 (2002) 185-220. | MR
[19] and , Exact and approximate controllability for distributed parameter systems. Acta Numerica (1996) 159-333. | MR
[20] , and , On the controllability of wave models with variable coefficients: a numerical investigation. Comput. Appl. Math. 21 (2002) 191-225. | MR
[21] , and , Exact and approximate controllability for distributed parameter systems: a numerical approach in vol. 117 of Encyclopedia Math. Appl. Cambridge University Press, Cambridge (2008). | MR
[22] and , Exact controllability of semi-linear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109-154. | MR
[23] , Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Recherches en Mathématiques Appliquées, Tomes 1 et 2. Masson. Paris (1988). | MR
[24] , A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377-418.
[25] , Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5 (2008) 331-351. | MR
[26] , A variational perspective on controllability. Inverse Problems 26 (2010) 015004. | MR
[27] , Optimal shape and position of the support of the internal exact control of a string. Systems Control Lett. 58 (2009) 136-140. | MR
[28] , Convex functions and duality in optimization problems and dynamics. In vol. II of Lect. Notes Oper. Res. Math. Ec. Springer, Berlin (1969). | MR
[29] , A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math. 52 (1973) 189-221. | MR
[30] , Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR
[31] , Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl. 75 (1996) 367-408. | MR
[32] , On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control. Optim. 37 (1999) 1568-1599. | MR
[33] , Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control. Optim. 39 (2000) 812-834. | MR
[34] , Propagation, observation, control and numerical approximations of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | MR
[35] , Control and numerical approximation of the wave and heat equations. In vol. III of Intern. Congress Math. Madrid, Spain (2006) 1389-1417. | MR
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