Precise estimates for biorthogonal families under asymptotic gap conditions

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell, based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behaviour: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and so to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results of Fattorini-Russell concerning biorthogonal families, using complex analysis techniques developed by Seidman, G\"uichal, Tenenbaum-Tucsnak, and Lissy.