This book gives a comprehensive treatment of the fundamental necessary and sufficient conditions for optimality for finite-dimensional, deterministic, optimal control problems. The emphasis is on the geometric aspects of the theory and on illustrating how these methods can be used to solve optimal control problems. It provides tools and techniques that go well beyond standard procedures and can be used to obtain a full understanding of the global structure of solutions for the underlying problem. The text includes a large number and variety of fully worked out examples that range from the classical problem of minimum surfaces of revolution to cancer treatment for novel therapy approaches. All these examples, in one way or the other, illustrate the power of geometric techniques and methods. The versatile text contains material on different levels ranging from the introductory and elementary to the advanced. Parts of the text can be viewed as a comprehensive textbook for both advanced undergraduate and all level graduate courses on optimal control in both mathematics and engineering departments. The text moves smoothly from the more introductory topics to those parts that are in a monograph style were advanced topics are presented. While the presentation is mathematically rigorous, it is carried out in a tutorial style that makes the text accessible to a wide audience of researchers and students from various fields, including the mathematical sciences and engineering. Heinz Schattler is an Associate Professor at Washington University in St. Louis in the Department of Electrical and Systems Engineering, Urszula Ledzewicz is a Distinguished Research Professor at Southern Illinois University Edwardsville in the Department of Mathematics and Statistics.

to Classical Complex Analysis Vol. 1 by Robert B. Burckel Kansas State University 1979 BIRKHAUSER VERLAG BASEL UND STUTTGART CIP-Kurztitelaufnahme der Deutschen Bibliothek Burckel, Robert B.: An introduction to classical complex analysis I by Robert B. Burckel. - Basel. Stuttgart: Birkhiiuser. Vol. I. - 1979. (Lehrbilcher und Monographien aus dem Gebiete der exakten Wissenschaften: Math. Reihe; Bd. 64) All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. (c) Birkhiiuser Verlag Basel, 1979 North and South America Edition published by ACADEMIC PRESS. INC. III Fifth Avenue, New York, New York 10003 (Pure and Applied Mathematics, A Series of Monographs and Textbooks, Volume 82) ISBN-13: 978-3-0348-9376-3 e-ISBN-13: 978-3-0348-9374-9 DOl: 10.1007/978-3-0348-9374-9 Library of Congress Catalog Card Number 78-67403 5 Contents Volume I PREFACE 9 Chapter 0 PREREQUISITES AND PRELIMINARIES 13 1 Set Theory 13 2 Algebra 14 3 The Battlefield 14 4 Metric Spaces 15 5 Limsup and All That 18 6 Continuous Functions 20 7 Calculus 21 Chapter I CURVES, CONNECTEDNESS AND CONVEXITY 22 1 Elementary Results on Connectedness 22 2 Connectedness of Intervals, Curves and Convex Sets 23 3 The Basic Connectedness Lemma 28 4 Components and Compact Exhaustions 29 5 Connectivity of a Set 33 6 Extension Theorems 37 Notes to Chapter I 39"

Based on a translation of the 6th edition of Gewöhnliche Differentialgleichungen by Wolfgang Walter, this edition includes additional treatments of important subjects not found in the German text as well as material that is seldom found in textbooks, such as new proofs for basic theorems. This unique feature of the book calls for a closer look at contents and methods with an emphasis on subjects outside the mainstream. Exercises, which range from routine to demanding, are dispersed throughout the text and some include an outline of the solution. Applications from mechanics to mathematical biology are included and solutions of selected exercises are found at the end of the book. It is suitable for mathematics, physics, and computer science graduate students to be used as collateral reading and as a reference source for mathematicians. Readers should have a sound knowledge of infinitesimal calculus and be familiar with basic notions from linear algebra; functional analysis is developed in the text when needed.

This special volume is dedicated to Boris M. Mordukhovich, on the occasion of his 60th birthday, and aims to celebrate his fundamental contributionsto variational analysis, generalizeddifferentiationand their applications.A main exampleof these contributions is Boris' recent opus magnus "Variational Analysis and Generalized Differentiation"(vols. I and II) [2,3]. A detailed explanationand careful description of Boris' research and achievements can be found in [1]. Boris' active work and jovial attitude have constantly inspired researchers of several generations, with whom he has generously shared his knowledgeand ent- siasm, along with his well-known warmth and human touch. Variationalanalysis is a rapidlygrowing?eld within pure and applied mathem- ics, with numerous applications to optimization, control theory, economics, en- neering, and other disciplines. Each of the 12 chapters of this volume is a carefully reviewed paper in the ?eld of variational analysis and related topics. Many chapters of this volume were presented at the International Symposium on Variational Analysis and Optimization (ISVAO), held in the Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, from November 28 to November 30, 2008. The symposium was organized in honour of Boris' 60thbirthday.It broughttogetherBorisandotherresearchersto discusssta- of-the-art results in variational analysis and its applications, with emphasis on op- mization and control. We thank the organizers and participants of the symposium, who made the symposium a highly bene?cial and enjoyable event. We are also grateful to all the authors of this special volume, who have taken the opportunityto celebrate Boris' birthdayand his decadesof contributionsto the area.

This edited monograph provides a comprehensive and in-depth analysis of sliding mode control, focusing on event-triggered implementation. The technique allows to prefix the steady-state bounds of the system, and this is independent of any boundary disturbances. The idea of event-triggered SMC is developed for both single input / single output and multi-input / multi-output linear systems. Moreover, the reader learns how to apply this method to nonlinear systems. The book primarily addresses research experts in the field of sliding mode control, but the book may also be beneficial for graduate students.

Boundary value problems which have variational expressions in form of inequal ities can be divided into two main classes. The class of boundary value prob lems (BVPs) leading to variational inequalities and the class of BVPs leading to hemivariational inequalities. The first class is related to convex energy functions and has being studied over the last forty years and the second class is related to nonconvex energy functions and has a shorter research "life" beginning with the works of the second author of the present book in the year 1981. Nevertheless a variety of important results have been produced within the framework of the theory of hemivariational inequalities and their numerical treatment, both in Mathematics and in Applied Sciences, especially in Engineering. It is worth noting that inequality problems, i. e. BVPs leading to variational or to hemivariational inequalities, have within a very short time had a remarkable and precipitate development in both Pure and Applied Mathematics, as well as in Mechanics and the Engineering Sciences, largely because of the possibility of applying and further developing new and efficient mathematical methods in this field, taken generally from convex and/or nonconvex Nonsmooth Analy sis. The evolution of these areas of Mathematics has facilitated the solution of many open questions in Applied Sciences generally, and also allowed the formu lation and the definitive mathematical and numerical study of new classes of interesting problems."

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

This book provides readers with a detailed insight into diverse and exciting recent developments in computational solid mechanics, documenting new perspectives and horizons. The topics addressed cover a wide range of current research, from computational materials modeling, including crystal plasticity, micro-structured materials, and biomaterials, to multi-scale simulations of multi-physics phenomena. Particular emphasis is placed on pioneering discretization methods for the solution of coupled non-linear problems at different length scales. The book, written by leading experts, reflects the remarkable advances that have been made in the field over the past decade and more, largely due to the development of a sound mathematical background and efficient computational strategies. The contents build upon the 2014 IUTAM symposium celebrating the 60th birthday of Professor Michael Ortiz, to whom this book is dedicated. His work has long been recognized as pioneering and is a continuing source of inspiration for many researchers. It is hoped that by providing a "taste" of the field of computational mechanics, the book will promote its popularity among the mechanics and physics communities.

In wntmg this monograph my aim has been to present a "geometric" approach to the structural synthesis of multivariable control systems that are linear, time-invariant and of finite dynamic order. The book is ad dressed to graduate students specializing in control, to engineering scientists involved in control systems research and development, and to mathemati cians interested in systems control theory. The label "geometric" in the title is applied for several reasons. First and obviously, the setting is linear state space and the mathematics chiefly linear algebra in abstract (geometric) style. The basic ideas are the familiar system concepts of controllability and observability, thought of as geometric prop erties of distinguished state subspaces. Indeed, the geometry was first brought in out of revulsion against the orgy of matrix manipulation which linear control theory mainly consisted of, around fifteen years ago. But secondly and of greater interest, the geometric setting rather quickly sug gested new methods of attacking synthesis which have proved to be intuitive and economical; they are also easily reduced to matrix arithmetic as soon as you want to compute. The essence of the "geometric" approach is just this: instead of looking directly for a feedback law (say u = Fx) which would solve your synthesis problem if a solution exists, first characterize solvability as a verifiable property of some constructible state subspace, say Y. Then, if all is well, you may calculate F from Y quite easily."

This is the first book focusing exclusively on fuzzy dual numbers. In addition to offering a concise guide to their properties, operations and applications, it discusses some of their advantages with regard to classical fuzzy numbers, and describes the most important operations together with a set of interesting applications in e.g. optimization, decision-making and system design. The book provides students, researchers and professionals the necessary theoretical background to apply this particular subset of fuzzy numbers to decision-making problems involving uncertainty. Further, it shows how to solve selected engineering and management problems and includes detailed numerical examples.

As any human activity needs goals, mathematical research needs problems -David Hilbert Mechanics is the paradise of mathematical sciences -Leonardo da Vinci Mechanics and mathematics have been complementary partners since Newton's time and the history of science shows much evidence of the ben eficial influence of these disciplines on each other. Driven by increasingly elaborate modern technological applications the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a du ality gap between the two partners, and this gap is growing wider. Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multi-disciplinary publications which fall into the two following complementary categories: 1. An annual book dedicated to the latest developments in mechanics and mathematics; 2. Monographs, advanced textbooks, handbooks, edited vol umes and selected conference proceedings. The AMMA annual book publishes invited and contributed compre hensive reviews, research and survey articles within the broad area of modern mechanics and applied mathematics. Mechanics is understood here in the most general sense of the word, and is taken to embrace relevant physical and biological phenomena involving electromagnetic, thermal and quantum effects and biomechanics, as well as general dy namical systems. Especially encouraged are articles on mathematical and computational models and methods based on mechanics and their interactions with other fields. All contributions will be reviewed so as to guarantee the highest possible scientific standards."

Hemivariational inequalities represent an important class of problems in nonsmooth and nonconvex mechanics. By means of them, problems with nonmonotone, possibly multivalued, constitutive laws can be formulated, mathematically analyzed and finally numerically solved. The present book gives a rigorous analysis of finite element approximation for a class of hemivariational inequalities of elliptic and parabolic type. Finite element models are described and their convergence properties are established. Discretized models are numerically treated as nonconvex and nonsmooth optimization problems. The book includes a comprehensive description of typical representants of nonsmooth optimization methods. Basic knowledge of finite element mathematics, functional and nonsmooth analysis is needed. The book is self-contained, and all necessary results from these disciplines are summarized in the introductory chapter. Audience: Engineers and applied mathematicians at universities and working in industry. Also graduate-level students in advanced nonlinear computational mechanics, mathematics of finite elements and approximation theory. Chapter 1 includes the necessary prerequisite materials.

This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

This book exposes a number of mathematical models for fracture of growing difficulty. All models are treated in a unified way, based on incremental energy minimization. They differ from each other by the assumptions made on the inelastic part of the total energy, here called the "cohesive energy". Each model describes a specific aspect of material response, and particular care is devoted to underline the correspondence of each model to the experiments. The content of the book is a re-elaboration of the lectures delivered at the First Sperlonga Summer School on Mechanics and Engineering Sciences in September 2011. In the year and a half elapsed after the course, the material has been revised and enriched with new and partially unpublished results. Significant additions have been introduced in the occasion of the course "The variational approach to fracture and other inelastic phenomena", delivered at SISSA, Trieste, in March 2013. The Notes reflect a research line carried on by the writer over the years, addressed to a comprehensive description of the many aspects of the phenomenon of fracture, and to its relations with other phenomena, such as the formation of microstructure and the changes in the material's strength induced by plasticity and damage. Reprinted from the Journal of Elasticity, volume 112, issue 1, 2013.

The numerous applications of optimal control theory have given an incentive to the development of approximate techniques aimed at the construction of control laws and the optimization of dynamical systems. These constructive approaches rely on small parameter methods (averaging, regular and singular perturbations), which are well-known and have been proven to be efficient in nonlinear mechanics and optimal control theory (maximum principle, variational calculus and dynamic programming). An essential feature of the procedures for solving optimal control problems consists in the necessity for dealing with two-point boundary-value problems for nonlinear and, as a rule, nonsmooth multi-dimensional sets of differential equations. This circumstance complicates direct applications of the above-mentioned perturbation methods which have been developed mostly for investigating initial-value (Cauchy) problems. There is now a need for a systematic presentation of constructive analytical per turbation methods relevant to optimal control problems for nonlinear systems. The purpose of this book is to meet this need in the English language scientific literature and to present consistently small parameter techniques relating to the constructive investigation of some classes of optimal control problems which often arise in prac tice. This book is based on a revised and modified version of the monograph: L. D. Akulenko "Asymptotic methods in optimal control." Moscow: Nauka, 366 p. (in Russian)."

From the reviews: "The account is quite detailed and is written in a manner that will appeal to analysts and numerical practitioners alike...they contain everything from rigorous proofs to tables of numerical calculations.... one of the strong features of these books...that they are designed not for the expert, but for those who whish to learn the subject matter starting from little or no background...there are numerous examples, and counter-examples, to back up the theory...To my knowledge, no other authors have given such a clear geometric account of convex analysis." "This innovative text is well written, copiously illustrated, and accessible to a wide audience"

This volume collects research papers addressing topical issues in economics and management with a particular focus on dynamic models which allow to analyze and foster the decision making of firms in dynamic complex environments. The scope of the contributions ranges from daily operational challenges firms face to strategic choices in dynamic industry environments and the analysis of optimal growth paths. The volume also highlights recent methodological developments in the areas of dynamic optimization, dynamic games and meta-heuristics, which help to improve our understanding of (optimal) decision making in a fast evolving economy.

This book offers a comprehensive collection of the most advanced numerical techniques for the efficient and effective solution of simulation and optimization problems governed by systems of time-dependent differential equations. The contributions present various approaches to time domain decomposition, focusing on multiple shooting and parareal algorithms. The range of topics covers theoretical analysis of the methods, as well as their algorithmic formulation and guidelines for practical implementation. Selected examples show that the discussed approaches are mandatory for the solution of challenging practical problems. The practicability and efficiency of the presented methods is illustrated by several case studies from fluid dynamics, data compression, image processing and computational biology, giving rise to possible new research topics. This volume, resulting from the workshop Multiple Shooting and Time Domain Decomposition Methods, held in Heidelberg in May 2013, will be of great interest to applied mathematicians, computer scientists and all scientists using mathematical methods.

This book contains the edited version of lectures and contributed papers presented at the NA TO ADV ANCED RESEARCH WORKSHOP ON TOPOLOGY OPTIMIZATION OF STRUCTURES, held at Hotel do Mar, Sesimbra, Portugal, 20 June to 26 June, 1992, and organised by the Mathematical Institutc, The Technical University of Denmark and bv CEMUL-Center of Mechanics and Materials of the Techmcal University of Lisbon, Fifty participants from' fourteen countries attended the workshop, This book is organiLed in ten parts, each one addressing a sub field of topology optimiIalion, its relations to materials modelling and its implementation: Part 1- Topology design of discrete structures. Part II - Discrete design and sclection problems. Part III - The homogenization method for topology design. Part IV - AlternatiYC methods for topology design of continuum structures. Part V - Boundary shape design meth(xis. Part VI- Rela\.ation and optimal shape design. Part VII - EtTectiw media theory and optimal design. Part VIII - Extending the scope of topology design. Part IX- Topology design in a computer-aided design environment .. Parl X- Aspects of topology design.

Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics making up the title of the book, reflect the two origins of the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books.

Variational principles have proven to be surprisingly fertile. For example, Fermat used variational methods to demonstrate that light follows the fastest route from one point to another, an idea which came to be a cornerstone of geometrical optics. This book explains variational principles and charts their use throughout modern physics. It examines the analytical mechanics of Lagrange and Hamilton, the basic tools of any physicist. The book also offers simple but rich first impressions of Einstein’s General Relativity, Feynman’s Quantum Mechanics, and more that reveal amazing interconnections between various fields of physics.

Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis. Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.

This book addresses optimization in robotics, in terms of both the configuration space and the metal structure of the robot arm itself; and discusses, describes and builds different types of heuristics and algorithms in MATLAB. In addition, the book includes a wealth of examples and exercises. In particular, it enables the reader to write a MATLAB code for all the related problems in robotics. The book also offers detailed descriptions of and builds from scratch several types of optimization algorithms using MATLAB and simplified methods, especially for inverse problems and avoiding singularities. Each chapter features examples and exercises to enhance the reader's comprehension. Accordingly, the book offers the reader a better understanding of robot analysis from an optimization standpoint.

It is with great pleasure that I accepted invitation of Adnan Ibrahimbegovic to write this preface, for this invitation gave me the privilege to be one of the ?rsttoreadhisbookandallowedmetoonceagainemphasizetheimportance for our discipline of solid mechanics, which is currently under considerable development, to produce the reference books suitable for students and all other researchers and engineers who wish to advance their knowledge on the subject. Thesolidmechanicshascloselyfollowedtheprogressincomputerscienceand is currently undergoing a true revolution where the numerical modelling and simulations are playing the central role. In the industrial environment, the 'virtual' (or the computing science) is present everywhere in the design and engineering procedures. I have a habit of saying that the solid mechanics has become the science of modelling and inthat respectexpanded beyondits t- ditional frontiers. Several facets of current developments have already been treated in di?erent works published within the series 'Studies in mechanics of materials and structures'; for example, modelling heterogeneous materials (Besson et al. ), fracture mechanics (Leblond), computational strategies and namely LATIN method (Ladev' eze), instability problems (NQ Son) and ve- ?cation of ?nite element method (Ladev' eze-Pelle). To these (French) books, one should also add the work of Lemaitre-Chaboche on nonlinear behavior of solid materials and of Batoz on ?nite element method.

This book includes a self-contained theory of inequality problems and their applications to unilateral mechanics. Fundamental theoretical results and related methods of analysis are discussed on various examples and applications in mechanics. The work can be seen as a book of applied nonlinear analysis entirely devoted to the study of inequality problems, i.e. variational inequalities and hemivariational inequalities in mathematical models and their corresponding applications to unilateral mechanics. It contains a systematic investigation of the interplay between theoretical results and concrete problems in mechanics. It is the first textbook including a comprehensive and systematic study of both elliptic, parabolic and hyperbolic inequality models, dynamical unilateral systems and unilateral eigenvalues problems. The book is self-contained and it offers, for the first time, the possibility to learn about inequality models and to acquire the essence of the theory in a relatively short time.

Audience: The book is suitable for researchers, and for doctoral and post-doctoral courses.