Viorel Barbu – författare

This volume is concerned with optimal control problems governed by ordinary differential systems and partial differential equations. The emphasis is on first-order necessary conditions of optimality and the construction of optimal controllers in feedback forms. These subjects are treated using some new concepts and techniques in modern optimization theory, such as Clarke's generalized gradient, Ekeland's variational principle, viscosity solution to the Hamilton-Jacobi equation, and smoothing processes for optimal control problems governed by variational inequalities. A substantial part of this book is devoted to applications and examples. A background in advanced calculus should enable readers to understand most of this book, including the statement of the Pontriagin maximum principle and many of the applications. This work should be of interest to graduate students in mathematics and engineering, and researchers in applied mathematics, control theory and systems theory.

This volume is concerned with a few basic results in the classical and modern theory of boundary value problems for elliptic, parabolic and wave equations. The emphasis is on understanding the main results of the field, along with a few classical and modern methods. Weak solutions to boundary value problems of parabolic and hyperbolic type are also included. Furthermore, topics such as the maximum principle and potentials are presented. The book can be recommended for a graduate level course, and should be of interest to researchers and graduate students whose work involves partial differential equations, mathematics of physics and of engineering, potential theory and calculus of variations.

Stabilization of Navier–Stokes Flows presents recent notable progress in the mathematical theory of stabilization of Newtonian fluid flows.

This text focuses on the qualitative aspects of deterministic and stochastic differential equations. Areas covered include: ordinary and partial differential systems; optimal control of deterministic and stochastic evolution equations; control theory of partial differential equations (PDE's); optimization methods in PDE's with numerous applications to mechanics and physics; inverse problems; stability theory; abstract optimization problems; calculus of variations; numerical treatment of solutions to differential equations and related optimization problems. These research fields are under very active development and the present volume should be of interest to students and researchers working in applied mathematics or in system engineering. This volume contains selected contributions presented during the International Working Conference on Analysis and Optimization of Differential Systems, which was sponsored by the International Federation for Information Processing (IFIP) and held in Constanta, Romania in September 2002.Among the aims of this conference was the creation of new international contacts and collaborations, taking advantage of the new developments in Eastern Europe, particularly in Romania. The conference benefited from the support of the European Union via the EURROMMAT programme.

In the last decades, functional methods played an increasing role in the qualita tive theory of partial differential equations. The spectral methods and theory of C 0 semigroups of linear operators as well as Leray-Schauder degree theory, ?xed point theorems, and theory of maximal monotone nonlinear operators are now essential functional tools for the treatment of linear and nonlinear boundary value problems associated with partial differential equations. An important step was the extension in the early seventies of the nonlinear dy namics of accretive (dissipative) type of the Hille-Yosida theory of C semigroups 0 of linear continuous operators. The main achievement was that the Cauchy problem associated with nonlinear m accretive operators in Banach spaces is well posed and the corresponding dynamic is expressed by the Peano exponential formula from ?nite dimensional theory. This fundamental result is the corner stone of the whole existence theory of nonlinear in?nite dynamics of dissipative type and its contri bution to the development of the modern theory of nonlinear partial differential equations cannot be underestimated.

Stabilization of Navier–Stokes Flows presents recent notable progress in the mathematical theory of stabilization of Newtonian fluid flows.

In the last decades, functional methods played an increasing role in the qualita tive theory of partial differential equations. The spectral methods and theory of C 0 semigroups of linear operators as well as Leray-Schauder degree theory, ?xed point theorems, and theory of maximal monotone nonlinear operators are now essential functional tools for the treatment of linear and nonlinear boundary value problems associated with partial differential equations. An important step was the extension in the early seventies of the nonlinear dy namics of accretive (dissipative) type of the Hille-Yosida theory of C semigroups 0 of linear continuous operators. The main achievement was that the Cauchy problem associated with nonlinear m accretive operators in Banach spaces is well posed and the corresponding dynamic is expressed by the Peano exponential formula from ?nite dimensional theory. This fundamental result is the corner stone of the whole existence theory of nonlinear in?nite dynamics of dissipative type and its contri bution to the development of the modern theory of nonlinear partial differential equations cannot be underestimated.

Analysis and Optimization of Differential Systems focuses on the qualitative aspects of deterministic and stochastic differential equations.

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

This book delves into a rigorous mathematical exploration of the well-posedness and long-time behavior of weak solutions to nonlinear Fokker-Planck equations, along with their implications in the theory of probabilistically weak solutions to McKean-Vlasov stochastic differential equations and the corresponding nonlinear Markov processes.

Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found.The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model".The book will be of interest to PhD students and researchers in mathematics, physics and biology.

This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner.Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations.Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

This book is concerned with nonlinear semigroups of contractions in Banach spaces and their application to the existence theory for differential equa­ tions associated with nonlinear dissipative operators. The study of nonlinear semi groups resulted from the examination of nonlinear parabolic equations and from various nonlinear boundary value problems. The first work done by Y. Komura stimulated much further work and interest in this subject. Thus a series of studies was begun and then continued by T. Kato, M. G. Crandall, A. Pazy, H. Brezis and others, who made important con­ tributions to the development of the theory. The theory as developed below is a generalisation of the Hille-Yosida theory for one-parameter semigroups of linear operators and is a collection of diversified results unified more or less loosely by their methods of approach. This theory is also closely related to the theory of nonlinear monotone operators. Of course not all aspects of this theory could be covered in our expo­ sition, and many important contributions to the subject have been excluded for the sake of brevity. We have attempted to present the basic results to the reader and to orient him toward some of the applications. This book is intended to be self-contained. The reader is assumed to have only a basic knowledge of functional analysis, function theory and partial differential equations. Some of the necessary prerequisites for the reading of this 'book are summarized, with or without proof, in Chapter I.

This volume is concerned with a few basic results in the classical and modern theory of boundary value problems for elliptic, parabolic and wave equations. The emphasis is on understanding the main results of the field, along with a few classical and modern methods. Weak solutions to boundary value problems of parabolic and hyperbolic type are also included. Furthermore, topics such as the maximum principle and potentials are presented. Audience: The book can be recommended for a graduate level course, and will be of interest to researchers and graduate students whose work involves partial differential equations, mathematics of physics and of engineering, potential theory and calculus of variations.

An updated and revised edition of the 1986 title Convexity and Optimization in Banach Spaces, this book provides a self-contained presentation of basic results of the theory of convex sets and functions in infinite-dimensional spaces.

This work is a revised and enlarged edition of a book with the same title published in Romanian by the Publishing House of the Romanian Academy in 1989. It grew out of lecture notes for a graduate course given by the author at the University if Ia~i and was initially intended for students and readers primarily interested in applications of optimal control of ordinary differential equations. In this vision the book had to contain an elementary description of the Pontryagin maximum principle and a large number of examples and applications from various fields of science. The evolution of control science in the last decades has shown that its meth­ ods and tools are drawn from a large spectrum of mathematical results which go beyond the classical theory of ordinary differential equations and real analy­ ses. Mathematical areas such as functional analysis, topology, partial differential equations and infinite dimensional dynamical systems, geometry, played and will continue to play an increasing role in the development of the control sciences. On the other hand, control problems is a rich source of deep mathematical problems. Any presentation of control theory which for the sake of accessibility ignores these facts is incomplete and unable to attain its goals. This is the reason we considered necessary to widen the initial perspective of the book and to include a rigorous mathematical treatment of optimal control theory of processes governed by ordi­ nary differential equations and some typical problems from theory of distributed parameter systems.

An updated and revised edition of the 1986 title Convexity and Optimization in Banach Spaces, this book provides a self-contained presentation of basic results of the theory of convex sets and functions in infinite-dimensional spaces. The main emphasis is on applications to convex optimization and convex optimal control problems in Banach spaces. A distinctive feature is a strong emphasis on the connection between theory and application. This edition has been updated to include new results pertaining to advanced concepts of subdifferential for convex functions and new duality results in convex programming. The last chapter, concerned with convex control problems, has been rewritten and completed with new research concerning boundary control systems, the dynamic programming equations in optimal control theory and periodic optimal control problems. Finally, the structure of the book has been modified to highlight the most recent progression in the field including fundamental results on the theory of infinite-dimensional convex analysis and includes helpful bibliographical notes at the end of each chapter.

This book is concerned with functional methods (nonlinear semigroups of contractions, nonlinear m-accretive operators and variational techniques) in the theory of nonlinear partial differential equations of elliptic and parabolic type. In particular, applications to the existence theory of nonlinear parabolic equations, nonlinear Fokker-Planck equations, phase transition and free boundary problems are presented in details. Emphasis is put on functional methods in partial differential equations (PDE) and less on specific results.