Stability of observations of partial differential equations under uncertain perturbations

University of Dubrovnik, Department of Electrical Engineering and Computing, Ćira Carića 4, 20 000 Dubrovnik, Croatia.
martin.lazar@unidu.hr

Received: 20 January 2015
Revised: 24 October 2016
Accepted: 28 November 2016

Abstract

We demonstrate the stability of observability estimates for solutions to wave and Schrödinger equations subjected to additive perturbations. This work generalises recent averaged observability/control results by allowing for systems consisting of operators of different types. We also consider the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. Our analysis relies on microlocal defect tools, in particular on standard H-measures when the main system dynamic is governed by the wave operator, and parabolic H-measures in the case of the Schrödinger operator.