March  2021, 10(1): 129-153. doi: 10.3934/eect.2020054

Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation

1. 

INRIA, Villiers-lès-Nancy, F-54600, France

2. 

GIREF, Département de mathématiques et statistique, Université Laval, Québec, G1V 0A6, Canada

* Corresponding author: ludovick.gagnon@inria.fr

Received  January 2020 Revised  February 2020 Published  May 2020

Fund Project: The research of José M. Urquiza is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)

We study the boundary observability of the 1-D homogeneous wave equation when using a Legendre-Galerkin semi-discretization method. It is already known that spurious high frequencies are responsible for its lack of uniformity with respect to the discretization parameter [4] which may prevent convergence in the approximation of the associated controllability problem. A classical remedy is to filter out the highest frequency components but this comes with a high computational cost in several space dimensions. We present here three remedies: a spectral filtering method, a mixed formulation (already used in the context of finite element method [14]) and a Nitsche's method. Our numerical results show that the uniform boundary observability inequalities are recovered. On the other hand, surprisingly, none of them seem to provide the trace (or direct) inequality uniformly, a property used to prove the convergence of the numerical controls [11]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.

Citation: Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054
References:
[1]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[2]

L. Bales and I. Lasiecka, Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous ${L}_2$ Dirichlet boundary data, Math. Comp., 64 (1995), 89-115.  doi: 10.2307/2153324.  Google Scholar

[3]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

T. Z. Boulmezaoud and J. M. Urquiza, On the eigenvalues of the spectral second order differentiation operator and application to the boundary observability of the wave equation, J. Sci. Comput., 31 (2007), 307-345.  doi: 10.1007/s10915-006-9106-8.  Google Scholar

[5]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.  doi: 10.1016/S0764-4442(97)80053-5.  Google Scholar

[6]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Scientific Computation, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numer. Math., 102 (2006), 413-462.  doi: 10.1007/s00211-005-0651-0.  Google Scholar

[8]

C. CastroS. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square, IMA J. Numer. Anal., 28 (2008), 186-214.  doi: 10.1093/imanum/drm012.  Google Scholar

[9]

T. Chen and B. Francis, Optimal Sampled-data Control Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1996.  Google Scholar

[10]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

S. Ervedoza and E. Zuazua, The wave equation: Control and numerics, in Control of Partial Differential Equations, Lecture Notes in Mathematics, 2048, Springer, Berlin, Heidelberg, 2012,245–340. doi: 10.1007/978-3-642-27893-8_5.  Google Scholar

[12]

S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-5808-1.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  doi: 10.1007/BF03167891.  Google Scholar

[16]

M. J. GroteA. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal., 44 (2006), 2408-2431.  doi: 10.1137/05063194X.  Google Scholar

[17]

P. Hansbo, Nitsche's method for interface problems in computational mechanics, GAMM-Mitt., 28 (2005), 183-206.  doi: 10.1002/gamm.201490018.  Google Scholar

[18]

J. S. Hesthaven and R. M. Kirby, Filtering in Legendre spectral methods, Math. Comp., 77 (2008), 1425-1452.  doi: 10.1090/S0025-5718-08-02110-8.  Google Scholar

[19]

J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the $1$-D wave equation, M2AN Math. Model. Numer. Anal., 33 (1999), 407-438.  doi: 10.1051/m2an:1999123.  Google Scholar

[20]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[21]

I. LasieckaJ.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_{2}(0, \, T;L_{2}(\Gamma))$-Dirichlet boundary terms, Appl. Math. Optim., 10 (1983), 275-286.  doi: 10.1007/BF01448390.  Google Scholar

[23]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A nonconservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.  Google Scholar

[24]

I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290.  doi: 10.1007/BF01448201.  Google Scholar

[25]

I. Lasiecka and R. Triggiani, Differential and algebraic riccati equations with applications to boundary/point control problems: Continuous theory and approximation theory, in Lecture Notes in Control and Information Sciences, 164, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0006880.  Google Scholar

[26]

J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, in Research in Applied Mathematics, 8, Masson, Paris, 1988.  Google Scholar

[27]

A. Marica and E. Zuazua, Symmetric Discontinuous Galerkin Methods for 1-D Waves. Fourier Analysis, Propagation, Observability and Applications, SpringerBriefs in Mathematics, Springer, New York, 2014. doi: 10.1007/978-1-4614-5811-1.  Google Scholar

[28]

M. Negreanu and E. Zuazua, Convergence of a multigrid method for the controllability of a 1-d wave equation, C. R. Math. Acad. Sci. Paris, 338 (2004), 413-418.  doi: 10.1016/j.crma.2003.11.032.  Google Scholar

[29]

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9-15.  doi: 10.1007/BF02995904.  Google Scholar

[30]

J. Shen, Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.  Google Scholar

[31]

R. Triggiani, Exact boundary controllability on ${L}_2({\Omega})\times {H}^{-1}({\Omega})$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277.  doi: 10.1007/BF01443625.  Google Scholar

[32]

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

[33]

H. Vandeven, On the eigenvalues of second-order spectral differentiation operators, Comput. Methods Appl. Mech. Engrg., 80 (1990), 313-318.  doi: 10.1016/0045-7825(90)90035-K.  Google Scholar

[34]

T. Warburton and J. S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), 2765-2773.  doi: 10.1016/S0045-7825(03)00294-9.  Google Scholar

[35]

E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563.  doi: 10.1016/S0021-7824(98)00008-7.  Google Scholar

[36]

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

show all references

References:
[1]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[2]

L. Bales and I. Lasiecka, Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous ${L}_2$ Dirichlet boundary data, Math. Comp., 64 (1995), 89-115.  doi: 10.2307/2153324.  Google Scholar

[3]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

T. Z. Boulmezaoud and J. M. Urquiza, On the eigenvalues of the spectral second order differentiation operator and application to the boundary observability of the wave equation, J. Sci. Comput., 31 (2007), 307-345.  doi: 10.1007/s10915-006-9106-8.  Google Scholar

[5]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.  doi: 10.1016/S0764-4442(97)80053-5.  Google Scholar

[6]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Scientific Computation, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numer. Math., 102 (2006), 413-462.  doi: 10.1007/s00211-005-0651-0.  Google Scholar

[8]

C. CastroS. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square, IMA J. Numer. Anal., 28 (2008), 186-214.  doi: 10.1093/imanum/drm012.  Google Scholar

[9]

T. Chen and B. Francis, Optimal Sampled-data Control Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1996.  Google Scholar

[10]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

S. Ervedoza and E. Zuazua, The wave equation: Control and numerics, in Control of Partial Differential Equations, Lecture Notes in Mathematics, 2048, Springer, Berlin, Heidelberg, 2012,245–340. doi: 10.1007/978-3-642-27893-8_5.  Google Scholar

[12]

S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-5808-1.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  doi: 10.1007/BF03167891.  Google Scholar

[16]

M. J. GroteA. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal., 44 (2006), 2408-2431.  doi: 10.1137/05063194X.  Google Scholar

[17]

P. Hansbo, Nitsche's method for interface problems in computational mechanics, GAMM-Mitt., 28 (2005), 183-206.  doi: 10.1002/gamm.201490018.  Google Scholar

[18]

J. S. Hesthaven and R. M. Kirby, Filtering in Legendre spectral methods, Math. Comp., 77 (2008), 1425-1452.  doi: 10.1090/S0025-5718-08-02110-8.  Google Scholar

[19]

J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the $1$-D wave equation, M2AN Math. Model. Numer. Anal., 33 (1999), 407-438.  doi: 10.1051/m2an:1999123.  Google Scholar

[20]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[21]

I. LasieckaJ.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_{2}(0, \, T;L_{2}(\Gamma))$-Dirichlet boundary terms, Appl. Math. Optim., 10 (1983), 275-286.  doi: 10.1007/BF01448390.  Google Scholar

[23]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A nonconservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.  Google Scholar

[24]

I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290.  doi: 10.1007/BF01448201.  Google Scholar

[25]

I. Lasiecka and R. Triggiani, Differential and algebraic riccati equations with applications to boundary/point control problems: Continuous theory and approximation theory, in Lecture Notes in Control and Information Sciences, 164, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0006880.  Google Scholar

[26]

J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, in Research in Applied Mathematics, 8, Masson, Paris, 1988.  Google Scholar

[27]

A. Marica and E. Zuazua, Symmetric Discontinuous Galerkin Methods for 1-D Waves. Fourier Analysis, Propagation, Observability and Applications, SpringerBriefs in Mathematics, Springer, New York, 2014. doi: 10.1007/978-1-4614-5811-1.  Google Scholar

[28]

M. Negreanu and E. Zuazua, Convergence of a multigrid method for the controllability of a 1-d wave equation, C. R. Math. Acad. Sci. Paris, 338 (2004), 413-418.  doi: 10.1016/j.crma.2003.11.032.  Google Scholar

[29]

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9-15.  doi: 10.1007/BF02995904.  Google Scholar

[30]

J. Shen, Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.  Google Scholar

[31]

R. Triggiani, Exact boundary controllability on ${L}_2({\Omega})\times {H}^{-1}({\Omega})$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277.  doi: 10.1007/BF01443625.  Google Scholar

[32]

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

[33]

H. Vandeven, On the eigenvalues of second-order spectral differentiation operators, Comput. Methods Appl. Mech. Engrg., 80 (1990), 313-318.  doi: 10.1016/0045-7825(90)90035-K.  Google Scholar

[34]

T. Warburton and J. S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), 2765-2773.  doi: 10.1016/S0045-7825(03)00294-9.  Google Scholar

[35]

E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563.  doi: 10.1016/S0021-7824(98)00008-7.  Google Scholar

[36]

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

Figure 1.  Behaviour of the square roots of the discrete eigenvalues and of the continuous ones for $ N = 40 $
Figure 2.  Behaviour of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) with $ T = 8 $
Figure 3.  Behaviour of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) with $ T = 8 $} after Fourier filtering of order $ M $, with $ M $ the integer part of $ \frac{2}{\pi}N $ \label{constfourier
Figure 4.  Graphs of Cesàro, Lanczos, raised cosine and sharpened raised cosine filters
Figure 5.  Graphs of Vandeven (left) and exponential (right) filters for different values of $ p $
Figure 6.  Values of $ c_{N, T} $ (left column) and $ C_{N, T} $ (right column) for different filters and $ T = 8 $. The exponential and Vandeven filters (bottom) are both of order $ p = 4 $
Figure 7.  Values of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) for the exponential filter with $ p = 2 $ and $ T = 8 $
Figure 8.  Values of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) associated to the mixed Legendre Galerkin formulation (23)-(24) with $ T = 8 $
Figure 9.  Values of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) with Nitsche's method, with $ \gamma = 0.8 $ and $ T = 8 $
Figure 10.  Values of $ c_{N, T} $ (left) and $ C_{N, T} $ (right) for Nitsche's method, when the term $ \gamma N^2u^N(1, t) $ is dropped in their definitions (27) and (28), with $ \gamma = 0.8 $ and $ T = 8 $
Figure 11.  Numerical control $ v^N(t) $ obtained with the Legendre Galerkin method (15) for $ N = 32 $ (top left) $ N = 64 $ (top right), $ N = 128 $ (bottom left) and $ N = 256 $ (bottom right). $ T = 8 $
Figure 12.  Numerical controls with $ N = 128 $ with (from top to bottom): the classical formulation (15) (left), Exponential filtering (with p = 6) (right), the mixed formulation (left) and Nitsche's method (right)
Figure 13.  Errors $ |u_0^N-u_0|_{H^1_0} $, $ \|u_1^N-u_1\|_{L^2} $ and $ \|v^N-v\|_{L^2(0, T)} $ for the classical Legendre Galerkin method, and the remedies studied here : exponential spectral filtering (with p = 6), the mixed formulation and Nitsche's method ($ \gamma = 1 $), for $ N = 32, \, 64, \, 128, \, 256 $
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