Abstract: I will present null-controllability results for underactuated linear parabolic-transport systems with constant coefficients on the one-dimensional torus, in collaboration with Armand Koenig. The system couples transport and parabolic components through zero- and first-order constant terms, with a possibly non-diagonalizable diffusion matrix. The distributed control acts through a constant matrix, so that the number of controls may be strictly smaller than the number of equations.
The goal is to understand how the transport minimal time, the parabolic smoothing, and the algebraic structure of the couplings interact. In small time, these systems are never controllable, while in large time null-controllability holds, for sufficiently regular initial data, if and only if a spectral Kalman rank condition is satisfied on every Fourier mode.
A significant part of the talk will be devoted to the proof strategy. Starting from a fully actuated problem with fictitious controls, one has to eliminate the extra controls and recover a control lying in the prescribed underactuated range. This is achieved through an algebraic solvability argument, performed mode by mode in Fourier variables. I will first explain this mechanism in finite dimension, where it gives a useful viewpoint on the Kalman rank condition, before showing how it enters the analysis of the parabolic-transport system.
