Abstract:

This lecture addresses a long standing open problem posed by the late Professor J.L.Lions. It concerns with the optimal time and uniform boundary controllability and observability of wave equations in highly oscillating medium. The identification of this control problem after homogenization was established 30 years by M.Avellaneda and myself.， and optimal time of control were obtained via Bardos-Lebour-Rauch. However, there are counter-examples for uniform boundary controls observed via various authors, particularly by Zuazua and Castro. Here I shall present a recent joint work with Zhongwei Shen in which we proved uniform boundary controls for frequencies below some threshold (based on the typical seize of microstructures). The related eigenvalues and eigenfunctions would be also discussed.