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

Nicolae Cîndea; Enrique Fernández-Cara; Arnaud Münch

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 1076-1108
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Cîndea, Nicolae, Fernández-Cara, Enrique, and Münch, Arnaud. "Numerical controllability of the wave equation through primal methods and Carleman estimates." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 1076-1108. <http://eudml.org/doc/272846>.

@article{Cîndea2013,
abstract = {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.},
author = {Cîndea, Nicolae, Fernández-Cara, Enrique, Münch, Arnaud},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {one-dimensional wave equation; null controllability; finite element methods; Carleman estimates; boundary null controls},
language = {eng},
number = {4},
pages = {1076-1108},
publisher = {EDP-Sciences},
title = {Numerical controllability of the wave equation through primal methods and Carleman estimates},
url = {http://eudml.org/doc/272846},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 4
SP - 1076
EP - 1108
AB - 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.
LA - eng
KW - one-dimensional wave equation; null controllability; finite element methods; Carleman estimates; boundary null controls
UR - http://eudml.org/doc/272846
ER -

References

top
  1. [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. MR2558639
  2. [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. Zbl0786.93009MR1178650
  3. [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. [4] L. Baudouin, M. de Buhan and S. Ervedoza, Global Carleman estimates for wave and applications. Preprint. Zbl1267.93027
  5. [5] L. Baudouin and S. Ervedoza, Convergence of an inverse problem for discrete wave equations. Preprint. 
  6. [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. Zbl1139.93005MR2387911
  7. [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. Zbl1231.35110MR2818882
  8. [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. Zbl1194.35268MR2486082
  9. [9] G. Lebeau and M. Nodet, Experimental study of the HUM control operator for linear waves. Experiment. Math.19 (2010) 93–120. Zbl1190.35011MR2649987
  10. [10] I. Ekeland and R. Temam, Convex analysis and variational problems, Classics in Applied Mathematics. Soc. Industr. Appl. Math. SIAM, Philadelphia 28 (1999). Zbl0939.49002MR1727362
  11. [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). MR3220862
  12. [12] E. Fernández-Cara and A. Münch, Strong convergent approximations of null controls for the heat equation. Séma Journal61 (2013) 49–78. Zbl1263.35121
  13. [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. Zbl1322.93019
  14. [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 I349 (2011) 867–871. Zbl1226.93026
  15. [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 Fields2 (2012) 217–246. Zbl1264.35260MR2991568
  16. [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. Zbl0862.49004MR1406566
  17. [17] X. Fu, J. Yong, and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim.46 (2007) 1578–1614. Zbl1143.93005MR2361985
  18. [18] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptotic Analysis32 (2002) 185–220. Zbl1050.35046MR1993649
  19. [19] R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numerica (1996) 159–333. Zbl0838.93014MR1352473
  20. [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. Zbl1125.35309MR2009952
  21. [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). Zbl1142.93002MR2404764
  22. [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. Zbl0729.93023MR1086465
  23. [23] J-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Recherches en Mathématiques Appliquées, Tomes 1 et 2. Masson. Paris (1988). MR963060
  24. [24] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377–418. 
  25. [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. Zbl1242.49091MR2478708
  26. [26] P. Pedregal, A variational perspective on controllability. Inverse Problems 26 (2010) 015004. Zbl1189.93023MR2575349
  27. [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. Zbl1155.49312MR2493611
  28. [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). Zbl0186.23901MR334940
  29. [29] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math.52 (1973) 189–221. Zbl0274.35041MR341256
  30. [30] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20 (1978) 639–739. Zbl0397.93001MR508380
  31. [31] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl.75 (1996) 367–408. Zbl0896.35023MR1411157
  32. [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. Zbl0951.35069MR1710233
  33. [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. Zbl0982.35059MR1786331
  34. [34] E. Zuazua, Propagation, observation, control and numerical approximations of waves approximated by finite difference methods. SIAM Rev.47 (2005) 197–243. Zbl1077.65095MR2179896
  35. [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. Zbl1108.93023MR2275734

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.