Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

Abstract

We study linear systems, described by operators A, B, C for which the state space X is a Banach space. We suppose that −A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss' conjecture. The crucial assumptions on A are boundedness of an H ∞ -calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L p (Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω. Mathematics Subject Classification (2000). Primary 93C05, 93C20, 47D06, 47A10; Secondary 47A60.

R-Boundedness of Smooth Operator-Valued Functions

“…See [29] for information on this space. We note that a version of the following Lemma is also in [15,Proposition 3.19], where it is instead assumed that Y has so-called property (α). Moreover, in [15,Remark 5.3 and Proof of Proposition 3.19] it is claimed that this assumption can be relaxed to non-trivial cotype, which is weaker than our assumption below.…”

“…We note that a version of the following Lemma is also in [15,Proposition 3.19], where it is instead assumed that Y has so-called property (α). Moreover, in [15,Remark 5.3 and Proof of Proposition 3.19] it is claimed that this assumption can be relaxed to non-trivial cotype, which is weaker than our assumption below. However, there seems to be a small confusion there: in [15,Remark 5.3] it is observed that non-trivial cotype suffices, thanks to a result in [17], if (a certain Hilbert space) H 1 = C, whereas in [15,Proposition 3.19] and the following lemma one has the dual situation: H 1 = H is a general Hilbert space and H 2 = C. Indeed one could deduce the following lemma by a standard duality argument from the result in [17], but since a self-contained argument is only slightly longer, we provide it for completeness: .…”

On the $\ell^s$-boundedness of a family of integral operators

“…[6,Section 6], [7,Proposition 3.3 and Theorem 4.12], [13], [16,Section 3], [19,Section 4], [21,Chapter 2]). However, it seems they are of a different nature and cannot be used to prove Theorems 1.1, 3.10 and Corollary 3.14.…”

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

“…However, unfortunately, it is hard to test the admissibility, not to mention the well-posedness and regularity of an infinite-dimensional linear system with unbounded control and observation. There are many papers devoted to discussion on the admissibility of control and observation operators, most of which are interested in proving or disproving Weiss' conjecture (see, e.g., [6,[12][13][14][15]25,29,34,35]). Here, we mention an important work due to Zwart [35]; he proved that the Weiss conjecture almost holds in Hilbert spaces and in the case that p = 2.…”