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. more precise, each initial data y 0 ∈ L 2 ( 0 , 1 ) of the 1 -d linear heat equation has a boundary control of the minimal L 2 ( 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 φ ̂ 0 , minimizing a suitable quadratic cost. Our aim is to study the relationship between the regularity of y 0 and that of φ ̂ 0 . We show that there are regular data y 0 for which the corresponding φ ̂ 0 are highly irregular, not belonging to any negative exponent Sobolev space. Moreover, the class of such initial data y 0 is dense in L 2 ( 0 , 1 ) . This explains the severe ill-posedness of the numerical algorithms developed for the approximation of the minimal L 2 ( 0 , T ) -norm control of y 0 based on the computation of φ ̂ 0 . The lack of polynomial convergence rates for Tychonoff regularization processes is a consequence of this phenomenon too.