Séance Séminaire

Séminaire ACSIOM

Contrôlabilité à zéro pour une équation de la chaleur semi-linéaire

The talk deals with the numerical computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained by Fernandez-Cara and Zuazua in 2000 via a fixed point reformulation; and also by Barbu in 2000. More precisely, Carleman estimates and Kakutani's theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. In this talk, we introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. We also formulate and apply a Newton-Raphson algorithm in this context. Several numerical experiments that make it possible to test and compare these methods are performed.