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

  • * Corresponding author: Franck Boyer

    * Corresponding author: Franck Boyer
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  • 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.

    Mathematics Subject Classification: Primary: 35K10, 65M06, 93B05.


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  • Figure 1.  Typical geometric situation

    Figure 2.  Grid geometry

    Figure 3.  Component $ {\alpha} $ (left) and $ {\beta} $ (right) of system (4) with no control

    Figure 4.  Component $ {\alpha} $ (left) and $ {\beta} $ (right) of system (4) with a boundary control

    Figure 5.  Norms of the components $ {\alpha} $, $ {\beta} $ of system (4) with and without control

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