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The Wave Equation: Control and Numerics

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Control of Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2048))

Abstract

In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation. We develop the so-called discrete approach. In other words, we analyze to which extent the semidiscrete or fully discrete dynamics arising when discretizing the wave equation by means of the most classical scheme of numerical analysis, shear the property of being controllable, uniformly with respect to the mesh-size parameters and if the corresponding controls converge to the continuous ones as the mesh-size tends to zero. We focus mainly on finite-difference approximation schemes for the one-dimensional constant coefficient wave equation. Using the well known equivalence of the control problem with the observation one, we analyze carefully the second one, which consists in determining the total energy of solutions out of partial measurements. We show how spectral analysis and the theory of non-harmonic Fourier series allows, first, to show that high frequency wave packets may behave in a pathological manner and, second, to design efficient filtering mechanisms. We also develop the multiplier approach that allows to provide energy identities relating the total energy of solutions and the energy concentrated on the boundary. These observability properties obtained after filtering, by duality, allow to build controls that, normally, do not control the full dynamics of the system but rather guarantee a relaxed controllability property. Despite of this they converge to the continuous ones. We also present a minor variant of the classical Hilbert Uniqueness Method allowing to build smooth controls for smooth data. This result plays a key role in the proof of the convergence rates of the discrete controls towards the continuous ones. These results are illustrated by means of several numerical experiments.

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Notes

  1. 1.

    Inequality (5.41) is just an example of a variety of similar observability problems: (a) one could observe the energy concentrated on the extreme x = 0 or in the two extremes x = 0 and 1 simultaneously; (b) the L 2(0, T)-norm of u x (1, t) could be replaced by some other norm; (c) one could also observe the energy concentrated in a subinterval (α, β) of (0, 1), etc.

  2. 2.

    Microlocal analysis deals, roughly speaking, with the possibility of localizing functions and its singularities not only in the physical space but also in the frequency domain. Localization in the frequency domain may be done according to the size of frequencies but also to sectors in the euclidean space in which they belong to. This allows introducing the notion of microlocal regularity; see, for instance, [48].

  3. 3.

    Note, however, that tangent rays may be diffractive or even enter the boundary. We refer to [6] for a deeper discussion of these issues.

  4. 4.

    We refer to Grisvard [45] for a discussion of these problems in the context of non-smooth domains.

  5. 5.

    Here and in what follows u N refers to the Nth component of the solution \(\vec{u}\) of the semidiscrete system, which obviously depends also on h.

  6. 6.

    This is a non generic fact that occurs only for the constant coefficient 1-d problem with uniform meshes.

  7. 7.

    Defining group velocity as the derivative of ω, i.e., of the curve in the dispersion diagram (see Fig. 5.5), is a natural consequence of the classical properties of the superposition of linear harmonic oscillators with close but not identical phases (see [21]). There is a one-to-one correspondence between the group velocity and the spectral gap which may be viewed as a discrete derivative of this diagram. In particular, when the group velocity decreases, the gap between consecutive eigenvalues also decreases.

  8. 8.

    Note that in Fig. 5.5, both for finite differences and elements, the semidiscrete and continuous curves are tangent at low frequencies. This is in agreement with the convergence property of the numerical scheme under consideration and with the fact that low-frequency wave packets travel essentially with the velocity of the continuous model.

  9. 9.

    For given initial data (y 0, y 1), the initial data for the controlled semidiscrete system (5.98) are taken to be approximations of (y 0, y 1) on the discrete mesh. The convergence of the controls \(\vec{{v}}_{h}\) in L 2(0, T) is then analyzed for the controls corresponding to these approximate initial data.

  10. 10.

    These results may also be obtained using discrete multiplier techniques (see [53] and [32] for an improved version with a sharp estimate of the time T(δ)).

  11. 11.

    This argument can be easily adapted to the case where the numerical approximation scheme is discrete in both space and time by taking discrete Fourier transforms in both variables.

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Acknowledgements

When preparing the last version of this manuscript we were supported by Alejandro Maas Jr., internship student from the Universidad Técnica Federico Santa María (UTFSM), Chile, visiting BCAM for two months early 2011. He contributed to improve our plots and also to run the numerical experiments we present here. We express our gratitude to him for his efficient and friendly help. This work was supported by the ERC Advanced Grant FP7–246775 NUMERIWAVES, the Grant PI2010–04 of the Basque Government, the ESF Research Networking Program OPTPDE and Grant MTM2008–03541 of the MICINN, Spain. The first author acknowledges the hospitality and support of the Basque Center for Applied Mathematics where part of this work was done.

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Authors and Affiliations

  1. CNRS; Institut de Mathématiques, de Toulouse UMR 5219, 31062, Toulouse, France

    Sylvain Ervedoza

  2. Université de Toulouse; UPS, INSA, INP, ISAE, UT1, UTM; IMT, 31062, Toulouse, France

    Sylvain Ervedoza

  3. BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo, 14, E-48009, Bilbao, Basque Country, Spain

    Enrique Zuazua

  4. Ikerbasque Research Professor, Ikerbasque - Basque Foundation for Science, E48011, Bilbao, Basque Country, Spain

    Enrique Zuazua

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  1. Sylvain Ervedoza
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  2. Enrique Zuazua
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Correspondence to Sylvain Ervedoza .

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Ervedoza, S., Zuazua, E. (2012). The Wave Equation: Control and Numerics. In: Control of Partial Differential Equations. Lecture Notes in Mathematics(), vol 2048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27893-8_5

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