Regularity issues for the null-controllability of the linear 1-d heat equation

The fact that the heat equation is controllable to zero in any bounded domain of the euclidean space, any time T > 0 and from any open subset of the boundary is well known. On the other hand, numerical experiments show the ill-posedness of the problem. In this paper we develop a rigorous analysis of the 1−d problem which provides a sharp description of this ill-posedness. To be more precise, to each initial data y0 ∈ L2(0, 1) of the 1−d linear heat equation it corresponds a boundary control of minimal L2(0, T)−norm which drives the state to zero in time T > 0. This control is given by a solution of the homogeneous adjoint equation with some initial data ϕ0, minimizing a suitable quadratic cost. Our aim is to study the relation between the regularity of y0 and that of ϕ0. We show that there are regular data y0 for which the corresponding ϕ0 are highly irregular, not belonging to any negative exponent Sobolev space. Moreover, the class of such initial data y0 is dense i...

• Type: Journal
• Year: 2011
• Where: SCL
• Authors: Sorin Micu, Enrique Zuazua
• Keywords: Ill Posedness, Image Analysis, Negative Exponent, Open Subset, SCL 2011