Internal controllability for parabolic systems involving analytic non-local terms

Pierre Lissy (1) , Enrique Zuazua (2, 3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Departamento de Matemáticas [Madrid]
3 LJLL - Laboratoire Jacques-Louis Lions

We deal with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable-a property that has been recently characterized in terms of a Kalman-like rank condition-, we give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. Our general result is illustrated by two simple examples.

Mots clés: parabolic systems; non-local potentials; null controllability; Kalman rank condition; analyticity

Domaines: Optimisation et contrôle [math.OC]

Pierre Lissy, Enrique Zuazua. Internal controllability for parabolic systems involving analytic non-local terms. Chinese Annals of Mathematics - Series B, 2018, Special issue in honor of the 80th birthday of Philippe G. Ciarler, 39 (2), pp.281-296. ⟨hal-01574999v2⟩