Controllability and observability for non-autonomous evolution equations: the averaged Hautus test

TL;DR

This paper introduces an averaged Hautus test to characterize observability in non-autonomous evolution systems, extending classical results to time-dependent operators and applying them to Schrödinger and damped wave equations.

Contribution

It develops an averaged Hautus condition for non-autonomous systems, providing a new criterion for exact observability in time-dependent operator settings.

Findings

Characterizes observability for skew-adjoint operators using the averaged Hautus test.

Extends the criterion to general operators with growth conditions.

Applies the results to Schrödinger and damped wave equations with time-dependent coefficients.

Abstract

We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schr\"odinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.