  In this work we present new results on the cost of the boundary
controllability of parabolic systems at time $T > 0$. In particular, we will
study optimal estimates of the control cost at time $T$ ($T$ small enough) when
the eigenvalues of the generator of the $C_0$ semigroup accumulate and do not
satisfy a gap condition. The main ingredient we will use is the moment method
combined with sharp estimates of the $L^2(0,T; \mathbb C)$-norm of the elements
of biorthogonal families to complex exponentials.
