The subject of the book is the "know-how" of applied mathematical modelling: how to construct specific models and adjust them to a new engineering environment or more precise realistic assumptions; how to analyze models for the purpose of investigating real life phenomena; and how the models can extend our knowledge about a specific engineering process.

Variational Inequalities and Frictional Contact Problems contains a carefully selected collection of results on elliptic and evolutionary quasi-variational inequalities including existence, uniqueness, regularity, dual formulations, numerical approximations and error estimates ones. By using a wide range of methods and arguments, the results are presented in a constructive way, with clarity and well justified proofs. This approach makes the subjects accessible to mathematicians and applied mathematicians. Moreover, this part of the book can be used as an excellent background for the investigation of more general classes of variational inequalities. The abstract variational inequalities considered in this book cover the variational formulations of many static and quasi-static contact problems. Based on these abstract results, in the last part of the book, certain static and quasi-static frictional contact problems in elasticity are studied in an almost exhaustive way.

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.

The papers published in this volume focus on some of the most recent devel- ments in complementarity theory, variational principles, stability theory of fu- tional equations, nonsmooth optimization, and various other important topics of nonlinear analysis and optimization. This volume was initially planned to celebrate Professor George Isac’s 70th birthday by bringing together research scientists from mathematical domains which have long bene ted from Isac’s active research passion. Unfortunately, George Isac passed away in February 2009 at the age of 69. George Isac received his Ph. D. in 1973 from the Institute of Mathematics of the Romanian Academy of Sciences. He made outstanding contributions in s- eral branches of pure and applied mathematics, including complementarity theory, variational inequalities, xed point theory, scalar and vector optimization, theory of cones, eigenvalue problems, convex analysis, variational principles and regulari- tion methods, as well as a number of other topics. In his long and outstanding career, he wrote more than 200 papers and 13 books. Professor Isac was an avid traveler who visited more than 70 universities around the globe and delivered approximately 180 research presentations. He also authored seven books on poetry. During his s- enti c career he collaborated with numerous mathematicians. His research papers contain very deep, original and beautiful results. Through his signi cant contri- tions, he earned a distinguished position and became an internationally renowned leading scholar in his research elds.

This self-contained volume surveys three decades of mathematical control theory and at the same time describes how the work of Roger Brockett shaped and influenced its development. Nine survey articles written by leading experts in the field, who have also been closely associated with Roger Brockett at various stages in his career, treat the subject cohesively and in depth. This volume will provide an important reference for graduate students and researchers,as well as for mathematicians, engineers and scientists whose work involves concepts and the language of control and systems theory.

The classical Pontryagin maximum principle (addressed to deterministic finite dimensional control systems) is one of the three milestones in modern control theory. The corresponding theory is by now well-developed in the deterministic infinite dimensional setting and for the stochastic differential equations. However, very little is known about the same problem but for controlled stochastic (infinite dimensional) evolution equations when the diffusion term contains the control variables and the control domains are allowed to be non-convex. Indeed, it is one of the longstanding unsolved problems in stochastic control theory to establish the Pontryagintype maximum principle for this kind of general control systems: this book aims to give a solution to this problem. This book will be useful for both beginners and experts who are interested in optimal control theory for stochastic evolution equations.

This book is devoted to the recent progress on the turnpike theory. The turnpike property was discovered by Paul A. Samuelson, who applied it to problems in mathematical economics in 1949. These properties were studied for optimal trajectories of models of economic dynamics determined by convex processes. In this monograph the author, a leading expert in modern turnpike theory, presents a number of results concerning the turnpike properties in the calculus of variations and optimal control which were obtained in the last ten years. These results show that the turnpike properties form a general phenomenon which holds for various classes of variational problems and optimal control problems. The book should help to correct the misapprehension that turnpike properties are only special features of some narrow classes of convex problems of mathematical economics.

Markov decision processes (MDPs), also called stochastic dynamic programming, were first studied in the 1960s. MDPs can be used to model and solve dynamic decision-making problems that are multi-period and occur in stochastic circumstances. There are three basic branches in MDPs: discrete-time MDPs, continuous-time MDPs and semi-Markov decision processes. Starting from these three branches, many generalized MDPs models have been applied to various practical problems. These models include partially observable MDPs, adaptive MDPs, MDPs in stochastic environments, and MDPs with multiple objectives, constraints or imprecise parameters.

A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way.

The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems. The model problem that is considered throughout the book is of the form (P) Min/(x) subjectto G(x) E K. xeX Here X and Y are Banach spaces, K is a closed convex subset of Y, and / : X -+ IR and G : X -+ Y are called the objective function and the constraint mapping, respectively.

When analyzing systems with a large number of parameters, the dimen­ sion of the original system may present insurmountable difficulties for the analysis. It may then be convenient to reformulate the original system in terms of substantially fewer aggregated variables, or macrovariables. In other words, an original system with an n-dimensional vector of states is reformulated as a system with a vector of dimension much less than n. The aggregated variables are either readily defined and processed, or the aggregated system may be considered as an approximate model for the orig­ inal system.

Artificial neural networks possess several properties that make them particularly attractive for applications to modelling and control of complex non-linear systems. Among these properties are their universal approximation ability, their parallel network structure and the availability of on- and off-line learning methods for the interconnection weights.

Control Theory for Linear Systems deals with the mathematical theory of feedback control of linear systems. It treats a wide range of control synthesis problems for linear state space systems with inputs and outputs. The book provides a treatment of these problems using state space methods, often with a geometric flavour.

Accurate models to describe real-world phenomena are indispensable for research in such scientific fields as physics, engineering, biology, chemistry, and economics. The tools and techniques of applied analysis facilitate the development of mathematical models and can thereby serve as an excellent resource for students and researchers in various scientific and mathematical disciplines.

This book explores the design of optimal trajectories for space maneuver vehicles (SMVs) using optimal control-based techniques. It begins with a comprehensive introduction to and overview of three main approaches to trajectory optimization, and subsequently focuses on the design of a novel hybrid optimization strategy that combines an initial guess generator with an improved gradient-based inner optimizer.