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Fredi Tröltzsch

Bio: Fredi Tröltzsch is an academic researcher from Technical University of Berlin. The author has contributed to research in topics: Optimal control & Pointwise. The author has an hindex of 36, co-authored 156 publications receiving 5247 citations. Previous affiliations of Fredi Tröltzsch include Chemnitz University of Technology & University of Hamburg.


Papers
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MonographDOI
20 Apr 2010
Abstract: Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces. The exposition begins with control problems with linear equation, quadratic cost function and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students. Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers. Alongside the main theme of the analysis of problems of optimal control, Troltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization.

1,290 citations

Book
20 Apr 2010
TL;DR: This chapter introduces the basic concepts of optimal control for linear elliptic partial differential equations and shows two different numerical approaches for control problems, based on the Galerkin finite element method.
Abstract: In this chapter we will introduce the basic concepts of optimal control for linear elliptic partial differential equations. At first we present the classical theory in functional spaces “à la J.L.Lions”, see [Lio71, Lio72]; then we will address the methodology based on the use of the Lagrangian functional (see, e.g., [Mau81, BKR00, Jam88]). Finally, we will show two different numerical approaches for control problems, based on the Galerkin finite element method.

415 citations

Journal ArticleDOI
TL;DR: The uniform convergence of discretized controls to optimal controls is proven under natural assumptions and error estimates for optimal controls in the maximum norm are estimated.
Abstract: We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.

275 citations

Journal ArticleDOI
TL;DR: An a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by parabolic and elliptic PDEs is deduced.
Abstract: The main focus of this paper is on an a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by parabolic and elliptic PDEs. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the POD model, is from the (unknown) exact one. Numerical examples illustrate the realization of the proposed approach for linear-quadratic problems governed by parabolic and elliptic partial differential equations.

147 citations


Cited by
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01 Jan 2009
TL;DR: In this paper, a criterion for the convergence of numerical solutions of Navier-Stokes equations in two dimensions under steady conditions is given, which applies to all cases, of steady viscous flow in 2D.
Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

1,568 citations

01 Jan 2016
TL;DR: The linear and nonlinear programming is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
Abstract: Thank you for downloading linear and nonlinear programming. As you may know, people have search numerous times for their favorite novels like this linear and nonlinear programming, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some infectious bugs inside their desktop computer. linear and nonlinear programming is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the linear and nonlinear programming is universally compatible with any devices to read.

943 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
28 Jan 2005
TL;DR: The Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K), thermal diffusivity: α, ≡ k/(ρ · Cp) (m /s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K).
Abstract: Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

636 citations