Optimization in elliptic problems with applications to mechanics of deformable bodies and fluid mechanics
type: Item
instance of: scholarly article
title: Optimization in elliptic problems with applications to mechanics of deformable bodies and fluid mechanics
zbMATH Open document ID: 0947.49001
author: W. G. Litvinov
published in: Operator Theory: Advances and Applications
publication date: 2000-07-18T00:00:00Z
review text: The monograph is devoted to mathematical models of optimal control problems from mechanics of deformable bodies (plates, shells,\dots) and fluid mechanics. Considerable attention is given to the mathematical setting of problems, to the relations between continuous models and discretized ones, to the dependence of solutions of the corresponding equations on controls, to the convergence of approximate solutions. A distinctive feature of the monograph is that mostly compact sets of control, for instance, bounded sets from Sobolev spaces embedded in Lebesgue spaces, are considered if controls appear in coefficients of elliptic operators or if the control is the shape of a body. The cause for that is the fact that in many cases smooth controls comply with the physical and engineering aspects of the problem. The monograph consists of six chapters. The Chapter 1 gives the functional-analytical framework of the book. The Chapters 2 and 3 are devoted to optimization problems for elliptic systems with controls in coefficients or in the right-hand side, among them various problems of eigenvalue or shape optimization are studied. In Chapter 4, direct problems for various mathematical models of plates and shells are investigated. The Chapter 5 is devoted to the optimization of structures (including some structures of composite materials), mostly the thickness of a plate or a shell being the control. In Chapter 6, various optimization problems for steady flows of viscous and nonlinear viscous fluids are studied, in particular, the optimization by body and surface forces, by the distribution of velocities on the inlet, by perforated walls, by the shape of domains or canals. The monograph is written in an accessible and self-contained manner. It would be of a great interest to researchers working in optimization theory for partial differential equations and its applications as well as for graduate students with interests in applications of the theory of optimal control to real prob…
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 74-02
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 1478508
zbMATH Keywords: elliptic equations
zbMATH Keywords: mechanics of deformable bodies
zbMATH Keywords: fluid mechanics
zbMATH Keywords: elliptic operators
zbMATH Keywords: shape optimization
zbMATH Keywords: optimal control
reviewed by: Uldis Raitums
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1478508