Sharp estimates of the one-dimensional boundary control cost for parabolic systems

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.