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doi: 10.3934/mcrf.2022028
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Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation

Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal, F-63000 Clermont-Ferrand, France

* Corresponding author: Arthur Bottois

Received  January 2022 Revised  May 2022 Early access June 2022

The aim of this paper is to study the boundary controllability of the linear elasticity system as a first-order system in both space and time. Using the observability inequality known for the usual second-order elasticity system, we deduce an equivalent observability inequality for the associated first-order system. Then, the control of minimal $ L^2 $-norm can be found as the solution to a space-time mixed formulation. This first-order framework is particularly interesting from a numerical perspective since it is possible to solve the space-time mixed formulation using only piecewise linear $ C^0 $-finite elements. Numerical simulations illustrate the theoretical results.

Citation: Arthur Bottois, Nicolae Cîndea. Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022028
References:
[1]

M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation—a numerical study, ESAIM Control Optim. Calc. Var., 3 (1998), 163-212.  doi: 10.1051/cocv:1998106.

[2]

E. BécacheP. Joly and C. Tsogka, A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal., 39 (2002), 2109-2132.  doi: 10.1137/S0036142999359189.

[3]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, volume 44 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36519-5.

[4]

E. BurmanA. FeizmohammadiA. Münch and L. Oksanen, Space time stabilized finite element methods for a unique continuation problem subject to the wave equation, ESAIM Math. Model. Numer. Anal., 55 (2021), S969-S991.  doi: 10.1051/m2an/2020062.

[5]

E. Burman, A. Feizmohammadi, A. Munch and L. Oksanen, Spacetime finite element methods for control problems subject to the wave equation, working paper or preprint, September 2021.

[6]

F. L. Cardoso-RibeiroD. Matignon and L. Lefèvre, A partitioned finite element method for power-preserving discretization of open systems of conservation laws, IMA J. Math. Control Inform., 38 (2021), 493-533.  doi: 10.1093/imamci/dnaa038.

[7]

N. CîndeaE. Fernández-Cara and A. Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, ESAIM Control Optim. Calc. Var., 19 (2013), 1076-1108.  doi: 10.1051/cocv/2013046.

[8]

N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations, Calcolo, 52 (2015), 245-288.  doi: 10.1007/s10092-014-0116-x.

[9]

R. Font and F. Periago, Numerical simulation of the boundary exact control for the system of linear elasticity, Appl. Math. Lett., 23 (2010), 1021-1026.  doi: 10.1016/j.aml.2010.04.030.

[10]

R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., 103 (1992), 189-221.  doi: 10.1016/0021-9991(92)90396-G.

[11]

R. GlowinskiW. Kinton and M. F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation, Internat. J. Numer. Methods Engrg., 27 (1989), 623-635.  doi: 10.1002/nme.1620270313.

[12]

R. GlowinskiC. H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: Description of the numerical methods, Japan J. Appl. Math., 7 (1990), 1-76. 

[13]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[14]

L. I. Ignat and E. Zuazua, Convergence of a two-grid algorithm for the control of the wave equation, J. Eur. Math. Soc. (JEMS), 11 (2009), 351-391.  doi: 10.4171/JEMS/153.

[15]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.

[16]

S. Montaner and A. Münch, Approximation of controls for linear wave equations: A first order mixed formulation, Math. Control Relat. Fields, 9 (2019), 729-758.  doi: 10.3934/mcrf.2019030.

[17]

A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control, SIAM J. Control Optim., 40 (2001), 777-800.  doi: 10.1137/S0363012998345615.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

A. Rincon and I.-S. Liu, On numerical approximation of an optimal control problem in linear elasticity, Divulg. Mat., 11 (2003), 91-107. 

[20]

A. SerhaniD. Matignon and G. Haine, Partitioned finite element method for port-Hamiltonian systems with boundary damping: anisotropic heterogeneous 2D wave equations, IFAC-PapersOnLine, 52 (2019), 96-101. 

[21] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second edition, 2005.  doi: 10.1017/CBO9780511755422.
[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.

show all references

References:
[1]

M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation—a numerical study, ESAIM Control Optim. Calc. Var., 3 (1998), 163-212.  doi: 10.1051/cocv:1998106.

[2]

E. BécacheP. Joly and C. Tsogka, A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal., 39 (2002), 2109-2132.  doi: 10.1137/S0036142999359189.

[3]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, volume 44 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36519-5.

[4]

E. BurmanA. FeizmohammadiA. Münch and L. Oksanen, Space time stabilized finite element methods for a unique continuation problem subject to the wave equation, ESAIM Math. Model. Numer. Anal., 55 (2021), S969-S991.  doi: 10.1051/m2an/2020062.

[5]

E. Burman, A. Feizmohammadi, A. Munch and L. Oksanen, Spacetime finite element methods for control problems subject to the wave equation, working paper or preprint, September 2021.

[6]

F. L. Cardoso-RibeiroD. Matignon and L. Lefèvre, A partitioned finite element method for power-preserving discretization of open systems of conservation laws, IMA J. Math. Control Inform., 38 (2021), 493-533.  doi: 10.1093/imamci/dnaa038.

[7]

N. CîndeaE. Fernández-Cara and A. Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, ESAIM Control Optim. Calc. Var., 19 (2013), 1076-1108.  doi: 10.1051/cocv/2013046.

[8]

N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations, Calcolo, 52 (2015), 245-288.  doi: 10.1007/s10092-014-0116-x.

[9]

R. Font and F. Periago, Numerical simulation of the boundary exact control for the system of linear elasticity, Appl. Math. Lett., 23 (2010), 1021-1026.  doi: 10.1016/j.aml.2010.04.030.

[10]

R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., 103 (1992), 189-221.  doi: 10.1016/0021-9991(92)90396-G.

[11]

R. GlowinskiW. Kinton and M. F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation, Internat. J. Numer. Methods Engrg., 27 (1989), 623-635.  doi: 10.1002/nme.1620270313.

[12]

R. GlowinskiC. H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: Description of the numerical methods, Japan J. Appl. Math., 7 (1990), 1-76. 

[13]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[14]

L. I. Ignat and E. Zuazua, Convergence of a two-grid algorithm for the control of the wave equation, J. Eur. Math. Soc. (JEMS), 11 (2009), 351-391.  doi: 10.4171/JEMS/153.

[15]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.

[16]

S. Montaner and A. Münch, Approximation of controls for linear wave equations: A first order mixed formulation, Math. Control Relat. Fields, 9 (2019), 729-758.  doi: 10.3934/mcrf.2019030.

[17]

A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control, SIAM J. Control Optim., 40 (2001), 777-800.  doi: 10.1137/S0363012998345615.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

A. Rincon and I.-S. Liu, On numerical approximation of an optimal control problem in linear elasticity, Divulg. Mat., 11 (2003), 91-107. 

[20]

A. SerhaniD. Matignon and G. Haine, Partitioned finite element method for port-Hamiltonian systems with boundary damping: anisotropic heterogeneous 2D wave equations, IFAC-PapersOnLine, 52 (2019), 96-101. 

[21] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second edition, 2005.  doi: 10.1017/CBO9780511755422.
[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.

Figure 1.  Domains $ Q $ and associated meshes. (a) A structured mesh of $ Q = (0, 1)^2 \times (0, T) $. (b) An example of non-convex domain $ \Omega $ and (c) a mesh of $ Q = \Omega \times (0, T) $ associated to this domain
Figure 2.  Initial datum $ (u^0, u^1) $ constructed in [12]
Figure 3.  Results for initial datum $ (u^0, u^1) $ displayed in Figure 2. (a) Evolution of the norm residuals $ ( \boldsymbol{g}_{n}, \boldsymbol{G}_{n}) $. (b) Norm $ L^{2} $ of the control $ \boldsymbol{h}(t) $
Figure 4.  Norm of the error between the exact and numerical controls for different values of $ h $ and two different values of $ \alpha_1 $
Figure 5.  Solution $ (\zeta_{n}, {\bf{\Theta}}_{n}) $ obtained once the conjugate algorithm converged for the mesh 5. (a) $ \zeta_{n} $. (b) $ \Theta_{n, 1} $. (c) $ \Theta_{n, 2} $
Figure 6.  Evolution with respect to the time $ t $ of the norms of primal and dual solutions for: (a) the wave equation; (b) the elasticity system and the initial data in Figure 2
Figure 7.  Norm of the control $ \boldsymbol{h} $ for the five meshes described in Table 1 computed from $ ( \boldsymbol{\zeta}_{n}, {\bf{\Theta}}_{n}) $ (left) and from $ ( \boldsymbol{w}_{n}, \boldsymbol{Q}_{n}) $ (right), for $ \alpha_{1} = 10^{-3} $ (up) and $ \alpha_{1} = 9\times 10^{-1} $ (bottom), respectively
Figure 8.  The six components of the solution for the initial datum in Figure 2 and the finest mesh in Table 1. (a) $ \zeta_{n, 1} $. (b) $ \Theta_{n, 11} $. (c) $ \Theta_{n, 12} $. (d) $ \zeta_{n, 2} $. (e) $ \Theta_{n, 21} $. (f) $ \Theta_{n, 22 } $
Figure 9.  Norm of the control for initial data given by (74). (a) $ \alpha_2 = 10^{-3} $ and different meshes. (b) Computation on the mesh $ \sharp 5 $ and different values for $ \alpha_2 $
Table 1.  Description of five meshes of the domain $ Q = (0, 1)^2 \times (0, T) $
Mesh number 1 2 3 4 5
Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
Number of nodes 3 267 23 814 76 880 179 867 345 933
Number of tetrahedra 15 600 127 200 426 600 1 017 600 1 980 000
Mesh number 1 2 3 4 5
Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
Number of nodes 3 267 23 814 76 880 179 867 345 933
Number of tetrahedra 15 600 127 200 426 600 1 017 600 1 980 000
Table 2.  Description of five meshes of the domain $ Q = \Omega \times (0, T) $ for $ \Omega $ displayed in Figure 1 (b)
Mesh number 1 2 3 4 5
Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
Number of nodes 4 557 29 707 99 094 212 234 406 945
Number of tetrahedra 21 510 155 700 515 080 1 185 760 2 303 550
Mesh number 1 2 3 4 5
Diameter $ h $ of elements $ \frac{1}{10} $ $ \frac{1}{20} $ $ \frac{1}{30} $ $ \frac{1}{40} $ $ \frac{1}{50} $
Number of nodes 4 557 29 707 99 094 212 234 406 945
Number of tetrahedra 21 510 155 700 515 080 1 185 760 2 303 550
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