# American Institute of Mathematical Sciences

June  2020, 10(2): 217-256. doi: 10.3934/mcrf.2019037

## Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries

 1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France 2 Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France

* Corresponding author: Franck Boyer

Received  July 2018 Published  June 2020 Early access  August 2019

The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than $1$. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general.

The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls.

To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.

Citation: Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control & Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037
##### References:

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##### References:
Typical geometric situation
Grid geometry
Component ${\alpha}$ (left) and ${\beta}$ (right) of system (4) with no control
Component ${\alpha}$ (left) and ${\beta}$ (right) of system (4) with a boundary control
Norms of the components ${\alpha}$, ${\beta}$ of system (4) with and without control
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