Observability from measurable sets for strongly coupled parabolic systems

Abstract

We characterize observability of strongly coupled parabolic systems from arbitrary space-time measurable sets of positive measure. For constant coercive diffusion matrices and arbitrary constant zeroth-order coupling and observation matrices, observability holds if and only if the generalized Kalman rank condition is satisfied at every Dirichlet eigenvalue. The diffusion matrix need not be diagonalizable, and the number of components is arbitrary. The observation is measured in $L^1$, yielding null controllability with bounded controls by duality. The proof combines a Remez-type inequality for exponential polynomials with a quantitative analysis of the mode-dependent Kalman condition. The resulting finite-dimensional observability estimate is uniform over the spatial modes up to a polynomial frequency loss. A spectral inequality and the Lebeau--Robbiano strategy then give observability of the full system.