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.