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

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 Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

“…Clearly, W (•, s)x s is the unique classical solution of (13). We conclude that W is generated by {A Q (t) | t ≥ 0} with regularity space {Ỹ t | t ≥ 0}.…”

“…In the literature most attention has been devoted to autonomous control systems. However, in view of applications, the interest in non-autonomous systems has been rapidly growing in recent years, see e.g., [22,13,27,7,31,19,6,18,30] and the references therein. In particular, a class of scattering passive linear non-autonomous linear systems of the forṁ…”

On the admissibility of observation operators in the context of maximal regularity

“…The estimate (11) seems to be a natural generalization of the estimate (1), see also [9,Definition 2.1]. On the other hand, one can see that…”

On the admissibility of observation operators for evolution families

“…We are interested in the concept of admissible observation operators for nonautonomous linear systems, see, e.g., [8,9,18]. Before going into details, let us first recall the definition of such operators in the autonomous case.…”