Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods

Abstract

This paper deals with the numerical computation of distributed null controls for semilinear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernandez-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani’s Theorem together ensure the existence of solutions to fixed points for an equivalent fixed point reformulated problem. A nontrivial difficulty appears when we want to extract from the associated Picard iterates a convergent (sub)sequence. In this paper, 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.