The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

This chapter is devoted to the proof of Theorem 1.1. The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem 

$$ \left\{ \begin{array}{l} ({\rm A - }\lambda {\rm )u = f }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm in D, } \\ {\rm Lu = }\mu {\rm (x')}\frac{{\partial {\rm u}}}{{\partial {\rm n}}} + \Upsilon (x')u = \varphi \,on\,\partial D \\ \end{array} \right. $$

(1.1)

to that of a first-order pseudo-differential operator T(λ) = LP(λ) on the boundary ∂D, just as in Section 4.3. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate 

$$\parallel u\parallel _{2,p} \le C(\lambda )(\parallel f\parallel _p + |\varphi| _{2 - 1/p,p} + \parallel u\parallel _p ).$$

(1.2)

Proof of Theorem 1.1

Homogenization in open sets with holes

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

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A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators

J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,—

H. Beckert

Papers

J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,—