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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. Bécache, P. 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. Burman, A. Feizmohammadi, A. 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-Ribeiro, D. 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. Cîndea, E. 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. Glowinski, W. 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. Glowinski, C. 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. Serhani, D. 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. Bécache, P. 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. Burman, A. Feizmohammadi, A. 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-Ribeiro, D. 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. Cîndea, E. 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. Glowinski, W. 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. Glowinski, C. 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. Serhani, D. 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.
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
Initial datum $(u^0, u^1)$ constructed in [12]
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 $\boldsymbol{h}(t)$
Norm of the error between the exact and numerical controls for different values of $h$ and two different values of $\alpha_1$
Solution $(\zeta_{n}, {\bf{\Theta}}_{n})$ obtained once the conjugate algorithm converged for the mesh 5. (a) $\zeta_{n}$. (b) $\Theta_{n, 1}$. (c) $\Theta_{n, 2}$
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
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
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) $\Theta_{n, 22 }$
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 $\alpha_2$
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
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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