OBSERVABILITY FOR NON-AUTONOMOUS SYSTEMS

We study non-autonomous observation systems \.x(t) = A(t)x(t), y(t) = C(t)x(t), x(0) = x(0) is an element of X, 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 subset of [0, T],T > 0. We present applications of the above result to families (A(t)) of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein-Uhlenbeck operators P(t) on Lp(Rd) with observation operators C(t)u = u|Omega(t). In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets (Omega(t)) such that the corresponding observation system satisfies a final-state observability estimate.