Controllability conditions and non-concentration phenomena for the heat and wave equations

The controllability problem for a given partial differential equation (PDE) consists in sending any initial condition to zero with a right-hand-side active only in a given subregion $\omega$. It is tightly connected to non-concentration properties for solutions of the said PDE: if no solution can concentrate outside of $\omega$, then controllability from $\omega$ holds, and vice versa. The talk will cover two examples of such phenomena.
First, I will explain how controllability of the heat equation is implied by so-called spectral estimates for frequency-localized functions. These spectral estimates are themselves equivalent to an equidistribution property of \(\omega\). Several examples will be given, including some original results on non-compact manifolds.
I will then talk about a similar problem for the damped wave equation. In that case, concentration of waves along geodesics of the manifold must be avoided to achieve controllability. When the damping is continuous, the Geometric Control Condition (GCC) gives a sharp condition on the control set \(\omega\): the damping must capture every geodesic in some finite time. I will present some generalizations of the GCC by Burq-Gérard and myself for discontinuous dampings on tori.