Numerical controllability of the wave equation through primal methods and Carleman estimates

Cîndea, Nicolae; Fernández-Cara, Enrique; Münch, Arnaud

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1076-1108.

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.

[1] M. Asch and A. Münch, 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] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR

[3] L. Baudouin, Lipschitz stability in an inverse problem for the wave equation, Master report (2001) available at: http://hal.archives-ouvertes.fr/hal-00598876/en/.

[4] L. Baudouin, M. De Buhan and S. Ervedoza, Global Carleman estimates for wave and applications. Preprint.

[5] L. Baudouin and S. Ervedoza, Convergence of an inverse problem for discrete wave equations. Preprint.

[6] C. Castro, S. Micu and A. Münch, 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] N. Cîndea, S. Micu and M. Tucsnak, An approximation method for the exact controls of vibrating systems. SIAM. J. Control. Optim. 49 (2011) 1283-1305. | MR

[8] B. Dehman and G. Lebeau, 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] G. Lebeau and M. Nodet, Experimental study of the HUM control operator for linear waves. Experiment. Math. 19 (2010) 93-120. | MR

[10] I. Ekeland and R. Temam, Convex analysis and variational problems, Classics in Applied Mathematics. Soc. Industr. Appl. Math. SIAM, Philadelphia 28 (1999). | MR

[11] S. Ervedoza and E. Zuazua, 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] E. Fernández-Cara and A. Münch, Strong convergent approximations of null controls for the heat equation. Séma Journal 61 (2013) 49-78.

[13] E. Fernández-Cara and A. Münch, 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] E. Fernández-Cara and A. Münch, 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] E. Fernández-Cara and A. Münch, 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] X. Fu, J. Yong, and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46 (2007) 1578-1614. | MR

[18] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptotic Analysis 32 (2002) 185-220. | MR

[19] R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numerica (1996) 159-333. | MR

[20] R. Glowinski, J. He and J.L. Lions, On the controllability of wave models with variable coefficients: a numerical investigation. Comput. Appl. Math. 21 (2002) 191-225. | MR

[21] R. Glowinski, J. He and J.L. Lions, 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] I. Lasiecka and R. Triggiani, Exact controllability of semi-linear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109-154. | MR

[23] J-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Recherches en Mathématiques Appliquées, Tomes 1 et 2. Masson. Paris (1988). | MR

[24] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377-418.

[25] A. Münch, 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] P. Pedregal, A variational perspective on controllability. Inverse Problems 26 (2010) 015004. | MR

[27] F. Periago, Optimal shape and position of the support of the internal exact control of a string. Systems Control Lett. 58 (2009) 136-140. | MR

[28] E.T. Rockafellar, Convex functions and duality in optimization problems and dynamics. In vol. II of Lect. Notes Oper. Res. Math. Ec. Springer, Berlin (1969). | MR

[29] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math. 52 (1973) 189-221. | MR

[30] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR

[31] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl. 75 (1996) 367-408. | MR

[32] P-F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control. Optim. 37 (1999) 1568-1599. | MR

[33] X. Zhang, 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] E. Zuazua, Propagation, observation, control and numerical approximations of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | MR

[35] E. Zuazua, Control and numerical approximation of the wave and heat equations. In vol. III of Intern. Congress Math. Madrid, Spain (2006) 1389-1417. | MR