Optimization Optimal Control And Partial Differential Equations

Author : Fredi Tröltzsch
Publisher : American Mathematical Society
Page : 417 pages
File Size : 46,6 Mb
Release : 2024-03-21
Category : Mathematics
ISBN : 9781470476441

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Optimal Control of Partial Differential Equations by Fredi Tröltzsch Pdf

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 equations, quadratic cost functions 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, Tröltzsch 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.

Author : Roland Herzog,Matthias Heinkenschloss,Dante Kalise,Georg Stadler,Emmanuel Trélat
Publisher : Walter de Gruyter GmbH & Co KG
Page : 386 pages
File Size : 45,9 Mb
Release : 2022-03-07
Category : Mathematics
ISBN : 9783110696004

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Optimization and Control for Partial Differential Equations by Roland Herzog,Matthias Heinkenschloss,Dante Kalise,Georg Stadler,Emmanuel Trélat Pdf

This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

Author : Roland Herzog,Matthias Heinkenschloss,Dante Kalise,Georg Stadler,Emmanuel Trélat
Publisher : Walter de Gruyter GmbH & Co KG
Page : 474 pages
File Size : 54,6 Mb
Release : 2022-03-07
Category : Mathematics
ISBN : 9783110695984

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Optimization and Control for Partial Differential Equations by Roland Herzog,Matthias Heinkenschloss,Dante Kalise,Georg Stadler,Emmanuel Trélat Pdf

This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

Author : Andrea Manzoni,Alfio Quarteroni,Sandro Salsa
Publisher : Springer Nature
Page : 507 pages
File Size : 44,8 Mb
Release : 2022-01-01
Category : Mathematics
ISBN : 9783030772260

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Optimal Control of Partial Differential Equations by Andrea Manzoni,Alfio Quarteroni,Sandro Salsa Pdf

This is a book on optimal control problems (OCPs) for partial differential equations (PDEs) that evolved from a series of courses taught by the authors in the last few years at Politecnico di Milano, both at the undergraduate and graduate levels. The book covers the whole range spanning from the setup and the rigorous theoretical analysis of OCPs, the derivation of the system of optimality conditions, the proposition of suitable numerical methods, their formulation, their analysis, including their application to a broad set of problems of practical relevance. The first introductory chapter addresses a handful of representative OCPs and presents an overview of the associated mathematical issues. The rest of the book is organized into three parts: part I provides preliminary concepts of OCPs for algebraic and dynamical systems; part II addresses OCPs involving linear PDEs (mostly elliptic and parabolic type) and quadratic cost functions; part III deals with more general classes of OCPs that stand behind the advanced applications mentioned above. Starting from simple problems that allow a “hands-on” treatment, the reader is progressively led to a general framework suitable to face a broader class of problems. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The three parts of the book are suitable to readers with variable mathematical backgrounds, from advanced undergraduate to Ph.D. levels and beyond. We believe that applied mathematicians, computational scientists, and engineers may find this book useful for a constructive approach toward the solution of OCPs in the context of complex applications.

Author : V. Barbu,J.F. Bonnans,D. Tiba
Publisher : Birkhäuser
Page : 344 pages
File Size : 53,7 Mb
Release : 2013-03-07
Category : Science
ISBN : 9783034886253

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Optimization, Optimal Control and Partial Differential Equations by V. Barbu,J.F. Bonnans,D. Tiba Pdf

This book collects research papers presented in the First Franco Romanian Conference on Optimization, Optimal Control and Partial Differential Equations held at lasi on 7-11 september 1992. The aim and the underlying idea of this conference was to take advantage of the new SOCial developments in East Europe and in particular in Romania to stimulate the scientific contacts and cooperation between French and Romanian mathematicians and teams working in the field of optimization and partial differential equations. This volume covers a large spectrum of problems and result developments in this field in which most of the participants have brought notable contributions. The following topics are discussed in the contributions presented in this volume. 1 -Variational methods in mechanics and physical models Here we mention the contributions of D. Cioranescu. P. Donato and H.I. Ene (fluid flows in dielectric porous media). R. Stavre (the impact of a jet with two fluids on a porous wall). C. Lefter and D. Motreanu (nonlinear eigenvalue problems with discontinuities). I. Rus (maximum principles for elliptic systems). and on asymptotic XII properties of solutions of evolution equations (R Latcu and M. Megan. R Luca and R Morozanu. R Faure). 2 -The controllabillty of Inflnlte dimensional and distributed parameter systems with the contribution of P. Grisvard (singularities and exact controllability for hyperbolic systems). G. Geymonat. P. Loreti and V. Valente (exact controllability of a shallow shell model). C.

Author : Fredi Tröltzsch
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 48,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821849040

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Optimal Control of Partial Differential Equations by Fredi Tröltzsch Pdf

Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. It includes topics on the existence of optimal solutions.

Author : Irwin Yousept
Publisher : Cuvillier Verlag
Page : 26 pages
File Size : 40,6 Mb
Release : 2008
Category : Electronic
ISBN : 9783867276689

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Optimal Control of Partial Differential Equations Involving Pointwise State Constraints: Regularization and Applications by Irwin Yousept Pdf

Author : Günter Leugering,Sebastian Engell,Andreas Griewank,Michael Hinze,Rolf Rannacher,Volker Schulz,Michael Ulbrich,Stefan Ulbrich
Publisher : Springer Science & Business Media
Page : 622 pages
File Size : 40,8 Mb
Release : 2012-01-03
Category : Mathematics
ISBN : 9783034801331

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Constrained Optimization and Optimal Control for Partial Differential Equations by Günter Leugering,Sebastian Engell,Andreas Griewank,Michael Hinze,Rolf Rannacher,Volker Schulz,Michael Ulbrich,Stefan Ulbrich Pdf

This special volume focuses on optimization and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since 2006. The book is organized in sections which cover almost the entire spectrum of modern research in this emerging field. Indeed, even though the field of optimal control and optimization for PDE-constrained problems has undergone a dramatic increase of interest during the last four decades, a full theory for nonlinear problems is still lacking. The contributions of this volume, some of which have the character of survey articles, therefore, aim at creating and developing further new ideas for optimization, control and corresponding numerical simulations of systems of possibly coupled nonlinear partial differential equations. The research conducted within this unique network of groups in more than fifteen German universities focuses on novel methods of optimization, control and identification for problems in infinite-dimensional spaces, shape and topology problems, model reduction and adaptivity, discretization concepts and important applications. Besides the theoretical interest, the most prominent question is about the effectiveness of model-based numerical optimization methods for PDEs versus a black-box approach that uses existing codes, often heuristic-based, for optimization.

Author : Jesús Martínez-Frutos,Francisco Periago Esparza
Publisher : Springer
Page : 123 pages
File Size : 51,6 Mb
Release : 2018-08-30
Category : Mathematics
ISBN : 9783319982106

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Optimal Control of PDEs under Uncertainty by Jesús Martínez-Frutos,Francisco Periago Esparza Pdf

This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition focusses on the entire path, starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail throughout the book. Computer codes, written in MatLab, are provided for all these examples. This book is adressed to graduate students and researches in Engineering, Physics and Mathematics who are interested in optimal control and optimal design for random partial differential equations.

Author : Michael Hinze,Rene Pinnau,Michael Ulbrich,Stefan Ulbrich
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 43,5 Mb
Release : 2008-10-16
Category : Mathematics
ISBN : 9781402088391

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Optimization with PDE Constraints by Michael Hinze,Rene Pinnau,Michael Ulbrich,Stefan Ulbrich Pdf

Solving optimization problems subject to constraints given in terms of partial d- ferential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of industrial, medical and economical applications, where the transition from model-based numerical si- lations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and num- ical simulation plays a central role. After proper discretization, the number of op- 3 10 timization variables varies between 10 and 10 . It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilize and f- ther explore the speci?c mathematical structure of optimization problems with PDE constraints, and to develop new mathematical approaches concerning mathematical analysis, structure exploiting algorithms, and discretization, with a special focus on prototype applications. The present book provides a modern introduction to the rapidly developing ma- ematical ?eld of optimization with PDE constraints. The ?rst chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in in?nite dim- sional spaces. Therefore, functional analytic techniques, function space theory, as well as existence- and uniqueness results for the underlying PDE are essential to study the existence of optimal solutions and to derive optimality conditions.

Author : Mieczyslaw Altman
Publisher : World Scientific
Page : 296 pages
File Size : 40,7 Mb
Release : 1990
Category : Science
ISBN : 9810203268

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A Theory of Optimization and Optimal Control for Nonlinear Evolution and Singular Equations by Mieczyslaw Altman Pdf

This research monograph offers a general theory which encompasses almost all known general theories in such a way that many practical applications can be obtained. It will be useful for mathematicians interested in the development of the abstract Control Theory with applications to Nonlinear PDE, as well as physicists, engineers, and economists looking for theoretical guidance in solving their optimal control problems; and graduate-level seminar courses in nonlinear applied functional analysis.

Author : José Luis Menaldi,Edmundo Rofman,Agnes Sulem
Publisher : IOS Press
Page : 632 pages
File Size : 46,7 Mb
Release : 2001
Category : Mathematics
ISBN : 1586030965

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Optimal Control and Partial Differential Equations by José Luis Menaldi,Edmundo Rofman,Agnes Sulem Pdf

This volume contains more than sixty invited papers of international wellknown scientists in the fields where Alain Bensoussan's contributions have been particularly important: filtering and control of stochastic systems, variationnal problems, applications to economy and finance, numerical analysis... In particular, the extended texts of the lectures of Professors Jens Frehse, Hitashi Ishii, Jacques-Louis Lions, Sanjoy Mitter, Umberto Mosco, Bernt Oksendal, George Papanicolaou, A. Shiryaev, given in the Conference held in Paris on December 4th, 2000 in honor of Professor Alain Bensoussan are included.

Author : Karl-Heinz Hoffmann,Günter Leugering,Fredi Tröltzsch
Publisher : Birkhäuser
Page : 324 pages
File Size : 48,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034886918

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Optimal Control of Partial Differential Equations by Karl-Heinz Hoffmann,Günter Leugering,Fredi Tröltzsch Pdf

The application of PDE-based control theory and the corresponding numerical algorithms to industrial problems have become increasingly important in recent years. This volume offers a wide spectrum of aspects of the discipline, and is of interest to mathematicians and scientists working in the field.

Author : Lorenz T. Biegler,Omar Ghattas,Matthias Heinkenschloss,David Keyes,Bart van Bloemen Waanders
Publisher : SIAM
Page : 335 pages
File Size : 41,8 Mb
Release : 2007-01-01
Category : Differential equations, Partial
ISBN : 0898718937

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Real-time PDE-constrained Optimization by Lorenz T. Biegler,Omar Ghattas,Matthias Heinkenschloss,David Keyes,Bart van Bloemen Waanders Pdf

Many engineering and scientific problems in design, control, and parameter estimation can be formulated as optimization problems that are governed by partial differential equations (PDEs). The complexities of the PDEs--and the requirement for rapid solution--pose significant difficulties. A particularly challenging class of PDE-constrained optimization problems is characterized by the need for real-time solution, i.e., in time scales that are sufficiently rapid to support simulation-based decision making. Real-Time PDE-Constrained Optimization, the first book devoted to real-time optimization for systems governed by PDEs, focuses on new formulations, methods, and algorithms needed to facilitate real-time, PDE-constrained optimization. In addition to presenting state-of-the-art algorithms and formulations, the text illustrates these algorithms with a diverse set of applications that includes problems in the areas of aerodynamics, biology, fluid dynamics, medicine, chemical processes, homeland security, and structural dynamics. Audience: readers who have expertise in simulation and are interested in incorporating optimization into their simulations, who have expertise in numerical optimization and are interested in adapting optimization methods to the class of infinite-dimensional simulation problems, or who have worked in "offline" optimization contexts and are interested in moving to "online" optimization.

Author : Peter I. Kogut,Günter R. Leugering
Publisher : Springer Science & Business Media
Page : 639 pages
File Size : 48,8 Mb
Release : 2011-09-09
Category : Science
ISBN : 9780817681494

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Optimal Control Problems for Partial Differential Equations on Reticulated Domains by Peter I. Kogut,Günter R. Leugering Pdf

In the development of optimal control, the complexity of the systems to which it is applied has increased significantly, becoming an issue in scientific computing. In order to carry out model-reduction on these systems, the authors of this work have developed a method based on asymptotic analysis. Moving from abstract explanations to examples and applications with a focus on structural network problems, they aim at combining techniques of homogenization and approximation. Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference tool for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.