Boundary null-controllability of some multi-dimensional linear parabolic systems by the moment method

Franck Boyer (1, 2) , Guillaume Olive (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IUF - Institut universitaire de France
3 Faculty of Mathematics and Computer Science of the Jagiellonian University

In this article we study the null controllability of some abstract linear parabolic systems in tensor product spaces. This special structure allows us to reduce our controllability problem to a particular set of equations that looks like a moment problem, but that does not fall into the previous existing results of the literature. We transform this non standard moment problem into an infinite family of more usual moment problems, yet coupled one from each other. This reformulation is done with enough care to ensure that the resulting set of equations can be solved, with suitable estimates, by using the recent "block moment method". This is based on a careful analysis of the spectral structure of the underlying operator. We notably apply our abstract result to show how strong the influence of geometry can be: we provide an example of boundary controlled parabolic system on a rectangle domain which is null controllable in arbitrary small time if two perpendicular faces of the boundary are controlled, whereas it is never null controllable if the control acts on only one face.

Mots clés: 2010 Mathematics Subject Classification. 93B05; 93C20; 93C25; 30E05; 35K90; 47A80

Domaines: Equations aux dérivées partielles [math.AP]

Franck Boyer, Guillaume Olive. Boundary null-controllability of some multi-dimensional linear parabolic systems by the moment method. 2021. ⟨hal-03175706v1⟩