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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Chapelle and K. J. Bathe, "The Finite Element Analysis of Shell - Fundamentals," Springer-Verlag, 2003.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[7]

Ph. Destuynder, A classification of thin shell theories, Act. Appl. Math., 1 (1985), 15-63. doi: 10.1007/BF02293490.  Google Scholar

[8]

Ph. Destuynder, "Modélisation des Coques Élastiques Minces," Masson, Paris, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," In Springer Series in Computational Physics, New York, 1984.  Google Scholar

[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.  Google Scholar

[14]

G. Grubb and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., 227 (1977), 247-276.  Google Scholar

[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.  Google Scholar

[16]

V. Komornik and P. Loreti, "Fourier Series in Control Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar

[17]

I. Lasiecka, R. Triggiani and V. Valente, Uniform stabilization of spherical shells by boundary dissipation, Advances in Differential Equations, 4 (1996), 635-674.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim, 35 (1997), 641-653. doi: 10.1137/S036301299526962X.  Google Scholar

[21]

J. L. Lions, "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués, Tome 1," Masson, RMA 8, Paris 1988.  Google Scholar

[22]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, (1968).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least square and variational approach,, Preprint. Available from , ().   Google Scholar

[27]

P. Pedregal, A variational perspective on controllability, Inverse Problems, 26 (2010), 015004. 17p. doi: 10.1088/0266-5611/26/1/015004.  Google Scholar

[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.  Google Scholar

[29]

J. Sanchez Hubert and E. Sanchez-Palencia, "Coques Élastiques Minces: Propriétés Asymptotiques," Masson, Paris, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Chapelle and K. J. Bathe, "The Finite Element Analysis of Shell - Fundamentals," Springer-Verlag, 2003.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[7]

Ph. Destuynder, A classification of thin shell theories, Act. Appl. Math., 1 (1985), 15-63. doi: 10.1007/BF02293490.  Google Scholar

[8]

Ph. Destuynder, "Modélisation des Coques Élastiques Minces," Masson, Paris, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," In Springer Series in Computational Physics, New York, 1984.  Google Scholar

[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.  Google Scholar

[14]

G. Grubb and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., 227 (1977), 247-276.  Google Scholar

[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.  Google Scholar

[16]

V. Komornik and P. Loreti, "Fourier Series in Control Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar

[17]

I. Lasiecka, R. Triggiani and V. Valente, Uniform stabilization of spherical shells by boundary dissipation, Advances in Differential Equations, 4 (1996), 635-674.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim, 35 (1997), 641-653. doi: 10.1137/S036301299526962X.  Google Scholar

[21]

J. L. Lions, "Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués, Tome 1," Masson, RMA 8, Paris 1988.  Google Scholar

[22]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, (1968).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least square and variational approach,, Preprint. Available from , ().   Google Scholar

[27]

P. Pedregal, A variational perspective on controllability, Inverse Problems, 26 (2010), 015004. 17p. doi: 10.1088/0266-5611/26/1/015004.  Google Scholar

[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.  Google Scholar

[29]

J. Sanchez Hubert and E. Sanchez-Palencia, "Coques Élastiques Minces: Propriétés Asymptotiques," Masson, Paris, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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