On the boundary controllability of non-scalar parabolic systems

Abstract

Abstract. This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. c 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Sur la contrôlabilité frontière des sytèmes paraboliques non scalaires Résumé. Cette Note concerne la contrôlabilité frontière des sytèmes paraboliques linéaires non scalaires. Plus précisement, on considère un système de deuxéquations paraboliques linéaires de dimension 1 en espace. Nous montrons qu'il est beaucoup plus compliqué de contrôler sur une partie du bord que de le faire avec des contrôles distribués. Dans notre résultat principal, on donne des conditions nécessaires et suffisantes pour la contrôlabilité exacteà zéro.

Boundary controllability of parabolic coupled equations

“…Remark 2.1. It is also well known that the approximate controllability of (1) can be characterized in terms of a property of the solutions to (10). More precisely, (1) is approximately controllable if and only if the following unique continuation property holds:…”

“…In the sequel, the solution to (10) will be called the adjoint state associated to ϕ 0 . The controllability of (1) can be characterized in terms of appropriate properties of the solutions to (10). More precisely, we have: Proposition 2.3.…”

“…Combining Proposition 2.3 and Theorem 1.1, we deduce the following:In the conditions of Theorem 1.1, there exists a positive constant C, only depending on T , such that the observability inequality(13) is satisfied by the solutions to(10) if and only if conditions(6) and(7)hold.…”

Null Controllability of a Parabolic System with a Cubic Coupling Term

“…Lemma 5 There exists a function G ∈ C ∞ ([0, +∞)) such that 17) and such that the solution g 0 to the Cauchy problem…”

“…See [15,20,22] for related results. See also [17] for boundary controls, [7] for some inverse problems and [18,23,13] for the Stokes system.…”