A note on an algorithm studying the uniform controllability of a class of semidiscrete hyperbolic problems
Abstract
We propose an algorithm based on the the technique introduced in [23]. The aim of the algorithm is to study, in a simple way, the approximation of the controls for a class of hyperbolic problems. It is well-known that, the finite-difference semi-discrete scheme for the approximation of controls can leads to high frequency numerical spurious oscillations which gives a loss of the uniform (with respect to the mesh-size) controllability property of the semidiscrete model. It is also known that an appropriate filtration of the high eigenfrequencies of the discrete initial data enable us to restore the uniform controllability property of the whole solution. But, the methods used to prove such results are very constructive and contains difficult and fine computations. As an example, which proves the effectiveness of our algorithm, we consider the case of the semidiscrete one dimensional wave equation. In this particular case, we are able to prove the uniform controllability, where the initial data are filtered in a range which contains as many modes as possibles, taking into account previous results obtained in
literature (see [18]).
Full Text:
PDFReferences
S.A. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.
U. Biccari, A. Marica, and E. Zuazua, Propagation of one- and two-dimensional discrete waves under finite difference approximation, Found. Comput. Math. 20 (2020), no. 6, 1401-1438.
I.F. Bugariu, S. Micu, and I. Rovența, Approximation of the controls for the beam equation with vanishing viscosity, Math. Comp. 85 (2016), 2259-2303.
I.F. Bugariu and I. Rovența, Small Time Uniform Controllability of the Linear One-Dimensional Schrodinger Equation with Vanishing Viscosity, J. Optim. Theory Appl. 160 (2014), no. 3, 949-965.
C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1d wave equation derived from a mixed finite element method, Numerische Mathematik 102 (2006), no. 3, 413-462.
N. Cindea, S. Micu, and I. Rovența, Boundary controllability for finite-differences semidiscretizations of a clamped beam equation, SIAM J. Control Optim. 55 (2017), 785-817.
N. Cindea, S. Micu, and M. Tucsnak, An approximation method for exact controls of vibrating systems, SIAM J. Control Optim. 49 (2011), no. 3, 1283-1305.
S. Ervedoza, Observability in arbitrary small time for discrete approximations of conservative systems, In: Some Problems on Nonlinear Hyperbolic Equations and Applications (T. Li, Y. Peng, B. Rao, eds.), Series in Contemporary Mathematics, Vol. 15, World Scientific, 2010, 283-309.
S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes, Numerische Mathematik 113 (2009), 377-415.
S. Ervedoza, Observability properties of a semi-discrete 1D wave equation derived from a mixed finite element method on nonuniform meshes, ESAIM Control Optim. Calc. Var. 16 (2010), 298-326.
S. Ervedoza, A. Marica, and E. Zuazua, Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis., IMA J. Numer. Anal. 36 (2016), 503-542.
S. Ervedoza and E. Zuazua, The wave equation: control and numerics, In: Control and stabilization of PDEs (P.M. Cannarsa, J.M. Coron, eds.), Lecture Notes in Mathematics 2048, CIME Subseries, Springer Verlag, 2012, 245-340.
S. Ervedoza and E. Zuazua, On the numerical approximation of exact controls for waves, Springer Briefs in Mathematics, Springer, New York, 2013.
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.
R. Glowinski, W. Kinton, and M.F. Wheeler, A mixed _nite element formulation for the boundary controllability of the wave equation, Internat. J. Numer. Methods Engrg. 27(1989), no. 3, 623-635.
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 University Press, New York, 2008.
J.A. Infante and E. Zuazua, Boundary observability for the space semi-discretization of the 1-D wave equation, Math. Model. Numer. Anal. 33 (1999), 407-438.
P. Lissy and I. Rovent_a, Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method, Math. Comp. 88 (2019), 273-291.
P. Lissy and I. Rovent_a, Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method, Mathematical Models and Methods in Applied Sciences 30 (2020), no. 3, 439-475.
S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation with vanishing viscosity, SIAM J. Cont. Optim. 47 (2008), 2857-2885.
S. Micu, I. Rovent_a, and L.E. Temereanc_a, Approximation of the controls for the wave equation with a potential, Numerische Mathematik 144 (2020), 835-887.
S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numerische Mathematik 91 (2002), 723-768.
S. Micu and I. Rovența, Filtration techniques for the uniform controllability of semi-discrete hyperbolic equations, Handbook of Numerical Analysis, Numerical Control: Part B 24 (2023), 261-296. DOI: 10.1016/bs.hna.2022.10.001
S. Micu and I. Rovența, Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity, ESAIM: COCV 18 (2012), 277-293.
S. Micu, I. Rovența, and L.E. Temereancă, Approximation of the controls for the linear beam equation, Math. Control Signals Systems 28 (2016), no. 2, Art. ID 12, 53 pp.
L. Miller, Resolvent conditions for the control of unitary groups and their approximations, Journal of Spectral Theory 2 (2012), 1-55.
A. Münch, Famille de schemas implicites uniformement controlables pour l'equation des ondes 1-D. (French.English, French summary) [Family of implicit schemes uniformly controllable for the 1-D wave equation], C. R. Math. Acad. Sci. Paris 339 (2004), no. 10, 733-738.
R.M. Redheffer, Completeness of sets of complex exponentials, Advances in Mathematics 24 (1977), 1-62.
D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review 20 (1978), 639-739.
R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference methods, SIAM Review 47 (2005), no. 2, 197-243.
DOI: https://doi.org/10.52846/ami.v50i1.1730