Null-controllability of linear parabolic-transport systems

Karine Beauchard (1) , Armand Koenig (2) , Kévin Le Balc’h (3)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 IRMAR - Institut de Recherche Mathématique de Rennes

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 proof of the negative result in small time uses holomorphic functions technics. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.

Mots clés: observability; parabolic-transport systems; null-controllability

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard, Armand Koenig, Kévin Le Balc’h. Null-controllability of linear parabolic-transport systems. 2019. ⟨hal-02191017v1⟩