Null-controllability of linear parabolic-transport systems

Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.