On the lack of null-controllability of the heat equation on the half-line

We consider the linear heat equation on the half-line with a Dirichlet boun dary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite tim e by means of an appropriate choice of the L-2 boundary control. We rewrite the system on the similarity variables that are a common tool when analyzi ng asymptotic problems. Next, the control problem is reduced to a moment pr oblem which turns out to be critical since it concerns the family of real e xponentials {e(jt)}(j greater than or equal to1) in which the usual summabi lity condition on the inverses of the eigenvalues does not hold. Roughly sp eaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the e xisting ones for bounded domains that guarantee that every initial datum be longing to a Sobolev space of negative order may be driven to zero in an ar bitrarily small time.