# American Institute of Mathematical Sciences

March  2013, 2(1): 119-151. doi: 10.3934/eect.2013.2.119

## A variational approach to approximate controls for system with essential spectrum: Application to membranal arch

 1 Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière

Received  September 2012 Revised  December 2012 Published  January 2013

We address the numerical approximation of boundary controls for systems of the form $\boldsymbol{y^{\prime\prime}}+\boldsymbol{A_M}\boldsymbol{y}=\boldsymbol{0}$ which models dynamical elastic shell structure. The membranal operator $\boldsymbol{A_M}$ is self-adjoint and of mixed order, so that it possesses a non empty and bounded essential spectrum $\sigma_{ess}(\boldsymbol{A_M})$. Consequently, the exact controllability does not hold uniformly with respect to the initial data. Thus the numerical computation of controls by the way of dual approach and gradient method may fail, even if the initial data belongs to the orthogonal of the space spanned by the eigenfunctions associated with $\sigma_{ess}(\boldsymbol{A_M})$. In this work, we adapt a variational approach introduced in [Pablo Pedregal, Inverse Problems (26) 015004 (2010)] for the wave equation and obtain a robust method of approximation. This new approach does not require any information on the spectrum of the operator $\boldsymbol{A_M}$. We also show that it allows to extract, from any initial data $(\boldsymbol{y^0},\boldsymbol{y^1})$, a controllable component for the mixed order system. We illustrate these properties with some numerical experiments. We also consider a relaxed controllability case for which uniform property holds.
Citation: Arnaud Münch. A variational approach to approximate controls for system with essential spectrum: Application to membranal arch. Evolution Equations and Control Theory, 2013, 2 (1) : 119-151. doi: 10.3934/eect.2013.2.119
##### References:
 [1] F. Ammar-Khodja, G. Geymonat and A. Münch, On the exact controllability of a system of mixed order with essential spectrum, C.R. Acad. Sci. Paris Série I, 346 (2008), 629-634. doi: 10.1016/j.crma.2008.04.009. [2] F. Ammar-Khodja, K. Mauffrey and A. Münch, Exact controllability of a system of mixed order with essential spectrum, SIAM J. of Control, 49 (2011), 1857-1879. doi: 10.1137/090777712. [3] 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. Numerical Analysis, 28 (2008), 186-214. doi: 10.1093/imanum/drm012. [4] D. Chapelle and K. J. Bathe, "The Finite Element Analysis of Shell - Fundamentals," Springer-Verlag, 2003. [5] P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells," In Studies in Mathematics and its Applications, 29., North-Holland Publishing Co., Amsterdam, 2000. [6] N. Cîndea, E. Fernández-Cara and A. Münch, Numerical controllability of the wave equation through a primal method and Carleman estimates, to appear in ESAIM: Control, Optimisation and Calculus of Variations. Available from http://hal.archives-ouvertes.fr/hal-00668951. [7] Ph. Destuynder, A classification of thin shell theories, Act. Appl. Math., 1 (1985), 15-63. doi: 10.1007/BF02293490. [8] Ph. Destuynder, "Modélisation des Coques Élastiques Minces," Masson, Paris, 1990. [9] G. Geymonat, P. Loreti and V. Valente, Contrôlabilité exacte d'un modèle de coque mince, C.R. Acad. Sci Série I, 313 (1991), 81-86. [10] G. Geymonat, P. Loreti and V. Valente, Exact controllability of thin elastic hemispherical shell via harmonic analysis, in "Boundary Value Problems for Partial Differential Equations and Applications" (eds. J. L. Lions and C. Baiocchi), Masson, (1993), 379-386. [11] G. Geymonat and V. Valente, A noncontrollability result for systems of mixed order, SIAM J. Control Optim, 39 (2000), 661-672. doi: 10.1137/S0363012998348322. [12] R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," In Springer Series in Computational Physics, New York, 1984. [13] R. Glowinski, J. L. Lions and J. He, "Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach," Encyclopedia of Mathematics and its Applications, 117, Cambridge, 2008. doi: 10.1017/CBO9780511721595. [14] G. Grubb and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., 227 (1977), 247-276. [15] G. Grubb and G. Geymonat, Eigenvalue asymptotics for selfadjoint elliptic mixed order systems with nonempty essential spectrum, Boll. Un. Mat. Ital., 16 (1979), 1032-1048. [16] V. Komornik and P. Loreti, "Fourier Series in Control Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. [17] I. Lasiecka, R. Triggiani and V. Valente, Uniform stabilization of spherical shells by boundary dissipation, Advances in Differential Equations, 4 (1996), 635-674. [18] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II. Abstract Hyperbolic-Like Systems Over a Finite Time Horizon," Encyclopedia of Mathematics and its Applications, 75. Cambridge University Press, Cambridge, 2000. [19] C. Lebiedzik, Exact boundary controllability of a shallow intrinsic shell model, J. Math. Anal. Appl., 335 (2007), 584-614. doi: 10.1016/j.jmaa.2007.01.061. [20] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim, 35 (1997), 641-653. doi: 10.1137/S036301299526962X. [21] J. L. Lions, "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués, Tome 1," Masson, RMA 8, Paris 1988. [22] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, (1968). [23] B. Miara and V. Valente, Exact controllability of a Koiter shell by a boundary action, J. Elasticity, 52 (1999), 267-287. doi: 10.1023/A:1007540610612. [24] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation, Mathematical Modeling and Numerical Analysis, 39 (2005), 377-418. doi: 10.1051/m2an:2005012. [25] A. Münch, Null boundary controllability of a circular elastic arch, IMA J. Mathematical Control and Information, 27 (2010), 119-144. doi: 10.1093/imamci/dnq004. [26] A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least square and variational approach, Preprint. Available from http://hal.archives-ouvertes.fr/hal-00724035 [27] P. Pedregal, A variational perspective on controllability, Inverse Problems, 26 (2010), 015004. 17p. doi: 10.1088/0266-5611/26/1/015004. [28] J. Sanchez Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Theory," Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-73782-4. [29] J. Sanchez Hubert and E. Sanchez-Palencia, "Coques Élastiques Minces: Propriétés Asymptotiques," Masson, Paris, 1993. [30] R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls, Control Cybernet., 25 (1996), 553-568. [31] L. Xiao, Asymptotic analysis of dynamic problems for linearly elastic shells - Justification of equations for dynamic Koiter shells, Chin. Ann. of Math., 22 (2001), 267-274. doi: 10.1142/S0252959901000279.

show all references

##### References:
 [1] F. Ammar-Khodja, G. Geymonat and A. Münch, On the exact controllability of a system of mixed order with essential spectrum, C.R. Acad. Sci. Paris Série I, 346 (2008), 629-634. doi: 10.1016/j.crma.2008.04.009. [2] F. Ammar-Khodja, K. Mauffrey and A. Münch, Exact controllability of a system of mixed order with essential spectrum, SIAM J. of Control, 49 (2011), 1857-1879. doi: 10.1137/090777712. [3] 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. Numerical Analysis, 28 (2008), 186-214. doi: 10.1093/imanum/drm012. [4] D. Chapelle and K. J. Bathe, "The Finite Element Analysis of Shell - Fundamentals," Springer-Verlag, 2003. [5] P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells," In Studies in Mathematics and its Applications, 29., North-Holland Publishing Co., Amsterdam, 2000. [6] N. Cîndea, E. Fernández-Cara and A. Münch, Numerical controllability of the wave equation through a primal method and Carleman estimates, to appear in ESAIM: Control, Optimisation and Calculus of Variations. Available from http://hal.archives-ouvertes.fr/hal-00668951. [7] Ph. Destuynder, A classification of thin shell theories, Act. Appl. Math., 1 (1985), 15-63. doi: 10.1007/BF02293490. [8] Ph. Destuynder, "Modélisation des Coques Élastiques Minces," Masson, Paris, 1990. [9] G. Geymonat, P. Loreti and V. Valente, Contrôlabilité exacte d'un modèle de coque mince, C.R. Acad. Sci Série I, 313 (1991), 81-86. [10] G. Geymonat, P. Loreti and V. Valente, Exact controllability of thin elastic hemispherical shell via harmonic analysis, in "Boundary Value Problems for Partial Differential Equations and Applications" (eds. J. L. Lions and C. Baiocchi), Masson, (1993), 379-386. [11] G. Geymonat and V. Valente, A noncontrollability result for systems of mixed order, SIAM J. Control Optim, 39 (2000), 661-672. doi: 10.1137/S0363012998348322. [12] R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," In Springer Series in Computational Physics, New York, 1984. [13] R. Glowinski, J. L. Lions and J. He, "Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach," Encyclopedia of Mathematics and its Applications, 117, Cambridge, 2008. doi: 10.1017/CBO9780511721595. [14] G. Grubb and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., 227 (1977), 247-276. [15] G. Grubb and G. Geymonat, Eigenvalue asymptotics for selfadjoint elliptic mixed order systems with nonempty essential spectrum, Boll. Un. Mat. Ital., 16 (1979), 1032-1048. [16] V. Komornik and P. Loreti, "Fourier Series in Control Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. [17] I. Lasiecka, R. Triggiani and V. Valente, Uniform stabilization of spherical shells by boundary dissipation, Advances in Differential Equations, 4 (1996), 635-674. [18] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II. Abstract Hyperbolic-Like Systems Over a Finite Time Horizon," Encyclopedia of Mathematics and its Applications, 75. Cambridge University Press, Cambridge, 2000. [19] C. Lebiedzik, Exact boundary controllability of a shallow intrinsic shell model, J. Math. Anal. Appl., 335 (2007), 584-614. doi: 10.1016/j.jmaa.2007.01.061. [20] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim, 35 (1997), 641-653. doi: 10.1137/S036301299526962X. [21] J. L. Lions, "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués, Tome 1," Masson, RMA 8, Paris 1988. [22] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, (1968). [23] B. Miara and V. Valente, Exact controllability of a Koiter shell by a boundary action, J. Elasticity, 52 (1999), 267-287. doi: 10.1023/A:1007540610612. [24] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation, Mathematical Modeling and Numerical Analysis, 39 (2005), 377-418. doi: 10.1051/m2an:2005012. [25] A. Münch, Null boundary controllability of a circular elastic arch, IMA J. Mathematical Control and Information, 27 (2010), 119-144. doi: 10.1093/imamci/dnq004. [26] A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least square and variational approach, Preprint. Available from http://hal.archives-ouvertes.fr/hal-00724035 [27] P. Pedregal, A variational perspective on controllability, Inverse Problems, 26 (2010), 015004. 17p. doi: 10.1088/0266-5611/26/1/015004. [28] J. Sanchez Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Theory," Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-73782-4. [29] J. Sanchez Hubert and E. Sanchez-Palencia, "Coques Élastiques Minces: Propriétés Asymptotiques," Masson, Paris, 1993. [30] R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls, Control Cybernet., 25 (1996), 553-568. [31] L. Xiao, Asymptotic analysis of dynamic problems for linearly elastic shells - Justification of equations for dynamic Koiter shells, Chin. Ann. of Math., 22 (2001), 267-274. doi: 10.1142/S0252959901000279.
 [1] Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603 [2] Monika Muszkieta. A variational approach to edge detection. Inverse Problems and Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009 [3] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [4] D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure and Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163 [5] Melek Jellouli. On the controllability of the BBM equation. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022002 [6] Venkatesan Govindaraj, Raju K. George. Controllability of fractional dynamical systems: A functional analytic approach. Mathematical Control and Related Fields, 2017, 7 (4) : 537-562. doi: 10.3934/mcrf.2017020 [7] Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206 [8] Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275 [9] Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233 [10] Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 [11] Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307 [12] Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925 [13] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 555-566. doi: 10.3934/naco.2020055 [14] Sylvain Ervedoza, Jérôme Lemoine, Arnaud Münch. Exact controllability of semilinear heat equations through a constructive approach. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022042 [15] Peng-Fei Yao. On shallow shell equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 697-722. doi: 10.3934/dcdss.2009.2.697 [16] Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291 [17] M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819 [18] Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems and Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443 [19] Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062 [20] Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164

2021 Impact Factor: 1.169