Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Approximation of the controls for the beam equation with vanishing viscosity


Authors: Ioan Florin Bugariu, Sorin Micu and Ionel Rovenţa
Journal: Math. Comp. 85 (2016), 2259-2303
MSC (2010): Primary 93B05, 58J45, 65N06, 30E05
DOI: https://doi.org/10.1090/mcom/3064
Published electronically: February 11, 2016
MathSciNet review: 3511282
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate boundary controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.


References [Enhancements On Off] (What's this?)

  • [1] Sergei A. Avdonin and Sergei A. Ivanov, Families of Exponentials, The method of moments in controllability problems for distributed parameter systems, Cambridge University Press, Cambridge, 1995. MR 1366650 (97b:93002)
  • [2] I. F. Bugariu and S. Micu, A numerical method for the controls of the heat equation, Math. Model. Nat. Phenom. 9 (2014), no. 4, 65-87. MR 3264295, https://doi.org/10.1051/mmnp/20149405
  • [3] C. Carthel, R. Glowinski, and J.-L. Lions, On exact and approximate boundary controllabilities for the heat equation: a numerical approach, J. Optim. Theory Appl. 82 (1994), no. 3, 429-484. MR 1290658 (95h:93008), https://doi.org/10.1007/BF02192213
  • [4] Jean-Michel Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. MR 2302744 (2008d:93001)
  • [5] Sylvain Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes, Numer. Math. 113 (2009), no. 3, 377-415. MR 2534130 (2010h:65153), https://doi.org/10.1007/s00211-009-0235-5
  • [6] Sylvain Ervedoza, Spectral conditions for admissibility and observability of Schrödinger systems: applications to finite element discretizations, Asymptot. Anal. 71 (2011), no. 1-2, 1-32. MR 2752768 (2012h:93031)
  • [7] Sylvain Ervedoza and Enrique Zuazua, Uniformly exponentially stable approximations for a class of damped systems, J. Math. Pures Appl. (9) 91 (2009), no. 1, 20-48. MR 2487899 (2010b:65131), https://doi.org/10.1016/j.matpur.2008.09.002
  • [8] Sylvain Ervedoza and Enrique Zuazua, A systematic method for building smooth controls for smooth data, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 4, 1375-1401. MR 2679646 (2011k:93036), https://doi.org/10.3934/dcdsb.2010.14.1375
  • [9] Sylvain Ervedoza and Enrique Zuazua, The wave equation: Control and numerics, Control of Partial Differential Equations, Lecture Notes in Math., vol. 2048, Springer, Heidelberg, 2012, pp. 245-339. MR 3220862, https://doi.org/10.1007/978-3-642-27893-8_5
  • [10] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272-292. MR 0335014 (48 #13332)
  • [11] H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math. 32 (1974/75), 45-69. MR 0510972 (58 #23325)
  • [12] R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys. 103 (1992), no. 2, 189-221. MR 1196839 (93j:65179), https://doi.org/10.1016/0021-9991(92)90396-G
  • [13] Roland Glowinski, Chin Hsien Li, and Jacques-Louis 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), no. 1, 1-76. MR 1039237 (91g:35043a), https://doi.org/10.1007/BF03167891
  • [14] R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 159-333. MR 1352473 (96m:93017)
  • [15] Scott W. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl. 158 (1991), no. 2, 487-508. MR 1117578 (92h:93055), https://doi.org/10.1016/0022-247X(91)90252-U
  • [16] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl. (9) 68 (1989), no. 4, 457-465 (1990). MR 1046761 (91e:93039)
  • [17] Thomas J. R. Hughes, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice Hall, Inc., Englewood Cliffs, NJ, 1987. MR 1008473 (90i:65001)
  • [18] Liviu I. Ignat and Enrique Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 47 (2009), no. 2, 1366-1390. MR 2485456 (2010c:35179), https://doi.org/10.1137/070683787
  • [19] A. E. Ingham, A Note on Fourier Transforms, J. London Math. Soc. S1-9, no. 1, 29. MR 1574706, https://doi.org/10.1112/jlms/s1-9.1.29
  • [20] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), no. 1, 367-379. MR 1545625, https://doi.org/10.1007/BF01180426
  • [21] E. Isaakson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, 1996.
  • [22] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugal. Math. 47 (1990), no. 4, 423-429. MR 1090480 (91j:93051)
  • [23] Vilmos Komornik and Paola Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. MR 2114325 (2006a:93001)
  • [24] Liliana León and Enrique Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation, ESAIM Control Optim. Calc. Var. 8 (2002), 827-862 (electronic). A tribute to J. L. Lions. MR 1932975 (2003k:93009), https://doi.org/10.1051/cocv:2002025
  • [25] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations]. MR 963060 (89k:93017)
  • [26] Sorin Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math. 91 (2002), no. 4, 723-768. MR 1912914 (2003f:93014), https://doi.org/10.1007/s002110100338
  • [27] Sorin Micu, Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity, SIAM J. Control Optim. 47 (2008), no. 6, 2857-2885. MR 2466095 (2011a:93019), https://doi.org/10.1137/070696933
  • [28] Luc Miller, Resolvent conditions for the control of unitary groups and their approximations, J. Spectr. Theory 2 (2012), no. 1, 1-55. MR 2879308, https://doi.org/10.4171/JST/20
  • [29] Arnaud Münch and Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, 265-293 (electronic). MR 2306636 (2008h:35217), https://doi.org/10.1051/cocv:2007009
  • [30] Raymond E. A. C. Paley and Norbert Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142 (98a:01023)
  • [31] Karim Ramdani, Takéo Takahashi, and Marius Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations--application to LQR problems, ESAIM Control Optim. Calc. Var. 13 (2007), no. 3, 503-527. MR 2329173 (2008g:34160), https://doi.org/10.1051/cocv:2007020
  • [32] Raymond M. Redheffer, Completeness of sets of complex exponentials, Advances in Math. 24 (1977), no. 1, 1-62. MR 0447542 (56 #5852)
  • [33] David L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), no. 4, 639-739. MR 508380 (80c:93032), https://doi.org/10.1137/1020095
  • [34] T. I. Seidman, S. A. Avdonin, and S. A. Ivanov, The ``window problem'' for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000), no. 3, 233-254. MR 1755142 (2001k:42038), https://doi.org/10.1007/BF02511154
  • [35] Louis Roder Tcheugoué Tébou and Enrique Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math. 95 (2003), no. 3, 563-598. MR 2012934 (2004m:65133), https://doi.org/10.1007/s00211-002-0442-9
  • [36] Louis T. Tebou and Enrique Zuazua, Uniform boundary stabilization of the finite difference space discretization of the $ 1$-$ d$ wave equation, Adv. Comput. Math. 26 (2007), no. 1-3, 337-365. MR 2350359 (2008j:93074), https://doi.org/10.1007/s10444-004-7208-0
  • [37] Marius Tucsnak and George Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. MR 2502023 (2010d:93001)
  • [38] Robert M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684 (81m:42027)
  • [39] Enrique Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47 (2005), no. 2, 197-243 (electronic). MR 2179896 (2006g:92017), https://doi.org/10.1137/S0036144503432862

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 93B05, 58J45, 65N06, 30E05

Retrieve articles in all journals with MSC (2010): 93B05, 58J45, 65N06, 30E05


Additional Information

Ioan Florin Bugariu
Affiliation: Department of Mathematics, University of Craiova, 200585, Romania
Email: florin$_$bugariu$_$86@yahoo.com

Sorin Micu
Affiliation: Department of Mathematics, University of Craiova, 200585 and Institute of Mathematical Statistics and Applied Mathematics, 70700, Bucharest, Romania
Email: sd$_$micu@yahoo.com

Ionel Rovenţa
Affiliation: Department of Mathematics, University of Craiova, 200585, Romania
Email: ionelroventa@yahoo.com

DOI: https://doi.org/10.1090/mcom/3064
Keywords: Beam equation, control approximation, vanishing viscosity, moment problem biorthogonals.
Received by editor(s): January 22, 2014
Received by editor(s) in revised form: September 11, 2014, and January 6, 2015
Published electronically: February 11, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society