This meeting brings together a few academics working in the theory of control and a few academics working in the theory of homogenization.

There are multiple themes for the discussion meeting.

Theme 1: This concerns the question raised by Jacques-Louis Lions in "Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1–68". This goes under the heading of Homogeization and Exact controllability in the context of wave equation discussed between pages 55 and 60 of the above review paper. The major breakthrough came in the following works by Carlos Castro and Enrique Zuazua:

• Castro, Carlos; Zuazua, Enrique, Controllability of the one-dimensional wave equation with rapidly oscillating density C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1237–1242.

• Castro, Carlos; Zuazua, Enrique Low frequency asymptotic analysis of a string with rapidly oscillating density, SIAM J. Appl. Math. 60 (2000), no. 4, 1205–1233.

• Castro, C.; Zuazua, E. High frequency asymptotic analysis of a string with rapidly oscillating density, European J. Appl. Math. 11 (2000), no. 6, 595–622. 

• Castro, C. Boundary controllability of the one-dimensional wave equation with rapidly oscillating density. Asymptot. Anal. 20 (1999), no. 3-4, 317–350.

The extension of the above one dimensional result to higher dimensions can be found in the work by G. Lebeau in "The wave equation with oscillating density: observability at low frequency, ESAIM Control Optim. Calc. Var. 5 (2000), 219–258". 

A recent work in this direction is by Fanghua Lin and Zhongwei Shen: "Uniform boundary controllability and homogenization of wave equations, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 9, 3031–3053". Their work relies on a homogenization result for eigenvalue problems obtained by C. Kenig, F. Lin and Z. Shen in "Estimates of eigenvalues and eigenfunctions in periodic homogenization, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1901–1925".

Theme 2: This concerns the necessary and sufficient conditions for the exact controllability of the wave equation. In particular, the Geometric Control Condition of Bardos, Lebeau and Rauch obtained in "Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065". This work relies on the microlocal analysis. 

Theme 3: This concerns the homogenization of optimal control problems. Here, the idea is to review a couple of results of Kesavan and co-authors and to understand a counterexample given by M. Avellaneda and F. Lin.

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