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

Abstract

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 {Λ k } k≥1 ⊂ 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 parabolic systems at time T > 0. In this case, the eigenvalues of the generator of the C 0 -semigroup associated to this parabolic system accumulate, do not satisfy a gap condition and can develop a positive minimal time for the null controllability.

Analysis of non scalar control problems for parabolic systems by the block moment method

“…Under a weak-gap condition of the form (23), that is when the eigenvalues can be gathered in blocks of bounded cardinality with a gap between blocks (which is the setting of the present article), uniform estimates on biorthogonal sequences follow from the uniform estimates for the resolution of block moment problems proved in [9]. Similar estimates, but where the sharp dependency with respect to T of the different constants is tracked, were obtained in [23]. Using the strategy detailed in [12], the estimates of [9] can also be supplemented with such dependency with respect to T (see Theorem 46).…”