Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{{\mathrm{\Lambda }}_k{\}}_{k\ge 1}\subset \mathbb{C}$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T>0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence.