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

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