Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

Abstract

We study linear control systems in infinite-dimensional Banach spaces governed by analytic semigroups. For p ∈ [1, ∞] and α ∈ R we introduce the notion of L p -admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first named author and Le Merdy [12] we give conditions under which L p -admissibility of type α is characterised by boundedness conditions which are similar to those in the well-known Weiss conjecture. We also study L p -wellposedness of type α for the full system. Here we use recent ideas due to Pruess and Simonett. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [4] to non-Hilbertian settings and to p = 2.

A direct approach to the Weiss conjecture for bounded analytic semigroups

“…Our approach does not make any recourse to the H ∞ -functional calculus and is based on elementary analysis. Similar results can be obtained for the weighted admissibility of observation operators studied in [9] and this will be the subject of a forthcoming paper.…”

“…As mentioned above this result was first obtained by Le Merdy [15] for p = 2 and by Haak-Kunstmann [9] for p ∈ (1, ∞]. The proof below includes the case p = 1.…”

“…In [5] this result was extended to the infinitetime p-admissibility. Recently, the authors in [9] have extended the result in [15] on the Weiss conjecture for 2-admissibility to the case of p-admissibility for bounded analytic semigroups.…”

“…The aim of this paper is to present new and much shorter proofs of the results of the Weiss-conjecture for analytic semigroups proved in [15] for p = 2 and in [9] for p ∈ (1, ∞], eventually even generalizing them. We will also prove that the analyticity assumption on the semigroup cannot be omitted.…”

On the Carleson Measure Criterion in Linear Systems Theory

“…For analytic semigroups, L 2 -admissibility is characterised by (5.2) even in Banach space under an inevitable additional condition (see [19]). This characterisation has subsequently been generalised to L p -norms with certain weights (see [13,12]). We briefly summarise some required notions and results:…”