On the regularity of null-controls of the linear 1-d heat equation

Abstract

The 1-d heat equation in a bounded interval is null-controllable from the boundary. More precisely, for each initial data y 0 ∈ L 2 ( 0 , 1 ) there corresponds a unique boundary control of minimal L 2 ( 0 , T ) -norm which drives the state of the 1-d linear heat equation to zero in time T > 0 . This control is given as the normal derivative of a solution of the homogeneous adjoint equation whose initial data φ ˆ 0 minimizes a suitable quadratic cost functional. In this Note we analyze the relation between the regularity of the initial datum to be controlled y 0 and that of φ ˆ 0 . We show that, if y 0 has only one Fourier mode, the corresponding φ ˆ 0 does not belong to any Sobolev space of negative exponent. This explains the severe ill-posedness of the problem and the lack of efficiency of most of the existing numerical algorithms for the numerical approximation of the controls, and in particular the slow convergence rate of Tychonoff regularization procedures.