Addendum to “W-stability and local input-output stability results”

In the above paper (Bourles and Colledani, 1995) a Sobolev space denoted as W was used in place of the well-known Lebesgue space L₂ for obtaining local input-output stability results, in particular a local version of the small gain theorem and of the passivity theorem. For stability studies, this space had to be extended. It was claimed that the signal space W has no classic extension. It is shown here that such an extension exists provided that the general notion of extended resolution space is used. This makes it possible to improve the results of Section II and to clarify the relations between the extended spaces