Volume 27, 2021
Special issue in the honor of Enrique Zuazua's 60th birthday
|Number of page(s)||26|
|Published online||22 June 2021|
Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation
Laboratory of Mathematics Informatics and Systems (LAMIS), University of Larbi Tebessi,
2 Departamento de Matemática e Informática, ETSI Caminos, Canales y Puertos, Polytecnical University of Madrid, 28040 Madrid, Spain.
* Corresponding author: firstname.lastname@example.org
Accepted: 31 May 2021
We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter σ. More precisely the control, independent of σ, that drives an initial data to a family of final states at time t = T, whose average in σ is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. When applied to a single (nonparametric) wave equation, the resulting discrete control problem turns out to be equivalent to the Galerkin approximation proposed by F. Bourquin et al. [C.R. Acad. Sci. Paris 313 I (1991) 757–760]. We give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain and allows us to give some conjectures on the averaged controllability for the continuous problem.
Mathematics Subject Classification: 35L05 / 65M70 / 65K10
Key words: Exact control / numerical approximation / averaged control / projection method
© The authors. Published by EDP Sciences, SMAI 2021
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