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H. Beckert

Bio: H. Beckert is an academic researcher. The author has an hindex of 1, co-authored 1 publications receiving 869 citations.


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Book
14 Jul 2009
TL;DR: The main topics of interest about observation and control operators are admissibility, observability, controllability, stabilizability and detectability as discussed by the authors, which is a mature area of functional analysis, which is still very active.
Abstract: 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

1,174 citations

Book ChapterDOI
01 Jan 1993
TL;DR: In this article, the authors describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions.
Abstract: 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.

704 citations

Book ChapterDOI
01 Jan 2004
TL;DR: Theorem 1.1.3 as discussed by the authors is proved in Section 4.3, and conditions (A and B) are sufficient for the validity of the a priori estimate.
Abstract: 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)

494 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of elastic torsion of a cylindrical bar with an increasing number of holes which are distributed periodically, and they proved that the problem is solvable using the energy method.

445 citations

Journal ArticleDOI
TL;DR: In this article, the authors show that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively, and they take this observation and weaken the smoothing assumptions on the operator and prove a novel convergence rate result.
Abstract: 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.

384 citations