Zero-class admissibility of observation operators

TL;DR

This paper investigates the conditions under which observation operators in control systems are zero-class admissible, establishing necessary and sufficient criteria, and illustrating the results with PDE examples.

Contribution

It develops new necessary and sufficient conditions for zero-class admissibility of observation operators, including a modified Weiss condition, and applies these to PDE systems.

Findings

Modified Weiss condition is necessary but not sufficient in general.

For certain C_0-semigroups, the modified Weiss condition is equivalent to zero-class admissibility.

Illustrative PDE examples demonstrate the theoretical results.

Abstract

An admissible observation operator is zero-class admissible if the norm of the output map tends to zero as the time tends to zero. Sufficient and necessary conditions for zero-class admissibility of observation operators are developed and a modified Weiss condition is studied. It is shown that the modified Weiss condition is in general necessary, but not sufficient for zero-class admissibility. For several important classes of C_0-semigroups it is proved that the modified Weiss condition is indeed equivalent to zero-class admissibility. The methods are illustrated by certain PDE examples.