A block moment method to handle spectral condensation phenomenon in parabolic control problems

This article is devoted to the characterization of the minimal null control time for abstract linear controlled problem. More precisely we aim at giving a precise answer to the following question: what is the minimal time needed to drive the solution of the system starting from any initial condition in a given subspace to zero ? Our setting will encompass a wide variety of systems of coupled one dimensional linear parabolic equations with a scalar control. Following classical ideas we reduce this controllability issue to the resolution of a moment problem on the control and provide a new block resolution technique for this moment problem. The obtained estimates are sharp and hold uniformly for a certain class of operators. This uniformity will allow various applications for parameter dependant control problems and permitted us to deal naturally with the case of algebraically multiple eigenvalues in the underlying generator. Our approach shed light on a new phenomenon: the condensation of eigenvalues (which can cause a non zero minimal null control time in general) can be somehow compensated by the condensation of eigenvectors. We provide various examples (some are abstract systems, others are actual PDE systems) to highlight those new situations we are able to manage by the block resolution of the moment problem we propose.