Observability for non-autonomous systems

Abstract

We study non-autonomous observation systems align* ẋ(t) = A(t) x(t), y(t) = C(t) x(t), x(0) = x₀∈ X, align* where (A(t)) is a strongly measurable family of closed operators on a Banach space X and (C(t)) is a family of bounded observation operators from X to a Banach space Y. Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in Lʳ(E; Y) for measurable subsets E ⊆ [0,T], T > 0. An application of the above result to families of uniformly strongly elliptic differential operators A(t) on Lᵖ(ℝᵈ) and observation operators C(t)u = 𝟏Ω₍t₎ u is presented. In this setting, we give sufficient and necessary geometric conditions on the family of sets (Ω(t)) such that the corresponding observation system satisfies a final-state observability estimate.