Observability for Non-autonomous Systems

Abstract

We study non-autonomous observation systems , where is a strongly measurable family of closed operators on a Banach space and is a family of bounded observation operators from to a Banach space . Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in for measurable subsets . We present applications of the above result to families of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein–Uhlenbeck operators on with observation operators . In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets such that the corresponding observation system satisfies a final-state observability estimate.