Abstract: In this talk, I will present some recent results on the bilinear control of some partial differential equations. Our geometric technique, based on "Lie bracket" arguments and actions of diffeomorphisms groups on densities, permit to give new information on (and in some cases completely characterize) the minimum time needed for global approximate controllability. We shall focus on Schrödinger and wave equations posed on tori or euclidean spaces, but such techniques actually apply to many other PDE systems posed on a general manifold with bilinear or linear structure in the control such as heat equation, transport equations along Hamiltonian fields, spin-coupled Schrödinger equations, and also transport equations appearing in deep learning architectures. A common hypothesis we made on those systems, is to consider controls supported only on a finite-dimensional subspace.
In particular, in the talk we will see that (i) for some Schrödinger equations the minimum control time for global L^2-approximate controllability is zero, independently of the initial state of the system, and (ii) for some wave equations the minimum control time for global H^1 x L^2-approximate controllability coincides with the maximum radius of a ball contained in the zero set of the initial state.
The strategy to proof (i) consists in decoupling the control of the Schrödinger equation into the control of two equations: an Hamilton-Jacobi equation for the angular phase, and a Liouville transport equation for the radial density, associated to the wavefunction. Our contribution is the control of the radial density thanks to a link with the control of the group of diffeomorphisms of the underlying torus or euclidean space. Such link is a consequence of a celebrated theorem of Jürgen Moser.
The talk will be based on the article [1] and the preprint [2] in collaboration with Karine Beauchard and Thomas Perrin (École Normale Supérieure de Rennes, France).
[1] Karine Beauchard, Eugenio Pozzoli; Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms. Ann. Inst. Henri Poincaré C Anal. Non Linéaire (2025), published online first.
[2] Karine Beauchard, Thomas Perrin, Eugenio Pozzoli; Approximate controllability of a bilinear wave equation and minimum time. (2026) Preprint on arXiv and HAL.
