On the reachable set for the one-dimensional heat equation

Jérémi Dardé (1) , Sylvain Ervedoza (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

The goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jérémi Dardé, Sylvain Ervedoza. On the reachable set for the one-dimensional heat equation. SIAM Journal on Control and Optimization, 2018, ⟨10.1137/16M1093215⟩. ⟨hal-01362214⟩