The flatness approach which is well-known for finite dimensional systems can be extended to some partial differential equations. In particular, it appears that this approach is a powerful toll to prove null controllability of the heat equation, with boundary control. This has been done in a series of articles by Martin, Rosier, Rouchon et al, see e.g. [4, 5].
In this talk, we will adapt the flatness method to tackle systems of heat equations. In particular, we will recover the controllability results given in Ammar-Khodja, Benabdallah, González-Burgos and De Teresa [1, 2].
This is an ongoing work with B. Colle and T. Takahashi, and the main matter of the presentation is taken from [3].

[1] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, and L. De Teresa. The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials. J. Math. Pures Appl. (9), 96(6):555–590, 2011.
[2] F. Ammar Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa. Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences. J. Funct. Anal., 267(7):2077–2151, 2014.
[3] B. Colle, J. Lohéac, and T. Takahashi. Flatness approach for the boundary controllability of a system of heat equations. Preprint, 2023.
[4] P. Martin, L. Rosier, and P. Rouchon. Null controllability of the heat equation using flatness. Automatica, 50(12):3067–3076, 2014.
[5] P. Martin, L. Rosier, and P. Rouchon. Null controllability of one-dimensional parabolic equations by the flatness approach. SIAM J. Control Optim., 54(1):198–220, 2016.