Numerical null controllability of the 1D heat equation: Carleman weights an duality

Enrique Fernandez-Cara (1) , Arnaud Munch (2)

1 Dpto. E.D.A.N. - Departamento de Ecuaciones Diferenciales y Análisis Numérico
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

This paper deals with the numerical computation of distributed null controls for the 1D heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state at t = 0 exactly to zero at t = T. We extend the earlier contribution of Carthel, Glowinski and Lions [5], which is devoted to the computation of minimal L2-norm controls. We start from some constrained extremal problems introduced by Fursikov and Imanuvilov [15]) and we apply appropriate duality techniques. Then, we introduce numerical approximations of the associated dual problems and we apply conjugate gradient algorithms. Finally, we present several experiments, we highlight the in uence of the weights and we analyze this approach in terms of robustness and e fficiency.

Mots clés: one-dimensional heat equation; null controllability; nite element methods; dual methods; conjugate gradient algorithm.; conjugate gradient algorithm

Domaines: Optimisation et contrôle [math.OC]

Enrique Fernandez-Cara, Arnaud Munch. Numerical null controllability of the 1D heat equation: Carleman weights an duality. 2011. ⟨hal-00687887v3⟩