How to understand Riemannian geometry is a necessary tool for control of the wave equation with variable coefficients

The exact controllability of the wave equation with variable coefficients had been a difficult topic for almost fifty years. There were many papers which changed the controllability into some uncheckable assumptions. The reason that these assumptions are uncheckable is because that controllability is a global property and the classical analysis works well for local problems only and is insufficient to cope with global problems. The differential geometrical approach was introduced more than a decade ago where the original motivation was to give checkable conditions to the exact controllability of the wave equation with variable coefficients. Since then, many important advances in modeling and control in vibrational and structural dynamics have been made. In this talk, we will briefly compare the Riemannian geometrical approach with some other main methods to show why it is a necessary tool to give checkable assumptions for controllability.