Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients

Abstract

In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms. Our analysis is mainly based on a criterion given by Fattorini in [12] (and systematically used in [22] for instance), that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems. This work gives theoretical justifications of some numerical observations described in [9].