Stability of Observations of Partial Differential Equations under Uncertain Perturbations

We analyse stability of observability estimates for solutions to wave and Scr\" odinger equations subjected to additive perturbations. The paper generalises the recent averaged observability/control result by allowing for systems consisting of operators of different types. The method also applies to the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. The analysis relies on microlocal defect tools; in particular on standard H-measures, when the main dynamic of the system is governed by the wave operator, while parabolic H-measures are explored in the case of the Schr\" odinger operator.