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Optimal Observation of the One-dimensional Wave Equation

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Abstract

In this paper, we consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0,π]. Let L∈(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0,π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L≠1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following Hébrard and Henrot (Syst. Control Lett., 48:199–209, 2003; SIAM J. Control Optim., 44:349–366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.

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Notes

  1. Here the uniqueness must be understood up to some subset of zero Lebesgue measure. In other words if ω is optimal then \(\omega\cup\mathcal{N}\) and \(\omega\setminus\mathcal{N}\) where \(\mathcal{N}\) denotes any subset of zero measure is also a solution.

  2. Note that, since the dynamics of (36) do not depend on the state, it follows that the adjoint states of the Pontryagin Maximum Principle are constant.

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Acknowledgements

The authors wish to thank Institut Henri Poincaré (Paris, France) for providing a very stimulating environment during the “Control of Partial and Differential Equations and Applications” program in the Fall 2010.

Y. Privat was partially supported by the ANR project GAOS “Geometric analysis of optimal shapes”.

E. Zuazua was partially supported by the Grant MTM2011-29306-C02-00 of the MICINN (Spain), project PI2010-04 of the Basque Government, the ERC Advanced Grant FP7-246775 NUMERIWAVES, the ESF Research Networking Program OPTPDE.

The authors warmly thank Aline Bonami, Giuseppe Buttazzo and Michel Crouzeix for useful discussions.

Author information

Authors and Affiliations

  1. IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170, Bruz, France

    Yannick Privat

  2. Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, Paris, France

    Emmanuel Trélat

  3. BCAM—Basque Center for Applied Mathematics, Mazarredo, 14, 48009, Bilbao, Spain

    Enrique Zuazua

  4. Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Spain

    Enrique Zuazua

Authors
  1. Yannick Privat
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  2. Emmanuel Trélat
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  3. Enrique Zuazua
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Corresponding author

Correspondence to Emmanuel Trélat.

Additional information

Communicated by Stéphane Jaffard.

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Privat, Y., Trélat, E. & Zuazua, E. Optimal Observation of the One-dimensional Wave Equation. J Fourier Anal Appl 19, 514–544 (2013). https://doi.org/10.1007/s00041-013-9267-4

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