With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

This IMA Volume in Mathematics and its Applications DISCRETE EVENT SYSTEMS, MANUFACTURING SYSTEMS AND COMMUNICATION NETWORKS is based on the proceedings of a workshop that was an integral part of the 1992-93 IMA program on "Control Theory. " The study of discrete event dynamical systems (DEDS) has become rapidly popular among researchers in systems and control, in communication networks, in manufacturing, and in distributed computing. This development has created problems for re searchers and potential "consumers" of the research. The first problem is the veritable Babel of languages, formalisms, and approaches, which makes it very difficult to determine the commonalities and distinctions among the competing schools of approaches. The second, related, problem arises from the different traditions, paradigms, values, and experience that scholars bring to their study of DEDS, depending on whether they come from control, com munication, computer science, or mathematical logic. As a result, intellectual exchange among scholars becomes compromised by unexplicated assumptions. The purpose of the Workshop was to promote exchange among scholars representing some of the major "schools" of thought in DEDS with the hope that (1) greater clarity will be achieved thereby, and (2) cross-fertilization will lead to more fruitful questions. We thank P. R. Kumar and P. P. Varaiya for organizing the workshop and editing the proceedings. We also take this opportunity to thank the National Science Foundation and the Army Research Office, whose financial support made the workshop possible. A vner Friedman Willard Miller, Jr."

This IMA Volume in Mathematics and its Applications VARIATIONAL AND FREE BOUNDARY PROBLEMS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries. " The aim of the workshop was to highlight new methods, directions and problems in variational and free boundary theory, with a concentration on novel applications of variational methods to applied problems. We thank R. Fosdick, M. E. Gurtin, W. -M. Ni and L. A. Peletier for organizing the year-long program and, especially, J. Sprock for co-organizing the meeting and co-editing these proceedings. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. Avner Friedman Willard Miller, Jr. PREFACE In a free boundary one seeks to find a solution u to a partial differential equation in a domain, a part r of its boundary of which is unknown. Thus both u and r must be determined. In addition to the standard boundary conditions on the un known domain, an additional condition must be prescribed on the free boundary. A classical example is the Stefan problem of melting of ice; here the temperature sat isfies the heat equation in the water region, and yet this region itself (or rather the ice-water interface) is unknown and must be determined together with the tempera ture within the water. Some free boundary problems lend themselves to variational formulation."

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modem mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises be covered in one class, are included in most chapters. Some units can whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.

The book is devoted to systems with discontinuous control. The study of discontinuous dynamic systems is a multifacet problem which embraces mathematical, control theoretic and application aspects. Times and again, this problem has been approached by mathematicians, physicists and engineers, each profession treating it from its own positions. Interestingly, the results obtained by specialists in different disciplines have almost always had a significant effect upon the development of the control theory. It suffices to mention works on the theory of oscillations of discontinuous nonlinear systems, mathematical studies in ordinary differential equations with discontinuous righthand parts or variational problems in nonclassic statements. The unremitting interest to discontinuous control systems enhanced by their effective application to solution of problems most diverse in their physical nature and functional purpose is, in the author's opinion, a cogent argument in favour of the importance of this area of studies. It seems a useful effort to consider, from a control theoretic viewpoint, the mathematical and application aspects of the theory of discontinuous dynamic systems and determine their place within the scope of the present-day control theory. The first attempt was made by the author in 1975-1976 in his course on "The Theory of Discontinuous Dynamic Systems" and "The Theory of Variable Structure Systems" read to post-graduates at the University of Illinois, USA, and then presented in 1978-1979 at the seminars held in the Laboratory of Systems with Discontinous Control at the Institute of Control Sciences in Moscow.

The author of this book made an attempt to create the general theory of optimization of linear systems (both distributed and lumped) with a singular control. The book touches upon a wide range of issues such as solvability of boundary values problems for partial differential equations with generalized right-hand sides, the existence of optimal controls, the necessary conditions of optimality, the controllability of systems, numerical methods of approximation of generalized solutions of initial boundary value problems with generalized data, and numerical methods for approximation of optimal controls. In particular, the problems of optimization of linear systems with lumped controls (pulse, point, pointwise, mobile and so on) are investigated in detail.

During the last half century, the development and testing of prediction models of combustion chamber performance have been an ongoing task at the International Flame Research Foundation (IFRF) in IJmuiden in the Netherlands and at many other research organizations. This task has brought forth a hierarchy of more or less standard numerical models for heat transfer predictions, in particular for the prediction of radiative heat transfer. Unfortunately all the methods developed, which certainly have a good physical foundation, are based on a large number of extreme sim plifications or uncontrolled assumptions. To date, the ever more stringent requirements for efficient production and use of energy and heat from com bustion chambers call for prediction algorithms of higher accuracy and more detailed radiative heat transfer calculations. The driving forces behind this are advanced technology requirements, the costs of large-scale experimen tal work, and the limitation of physical modeling. This interest is growing more acute and has increased the need for the publication of a textbook for more accurate treatment of radiative transfer in enclosures. The writing of a textbook on radiative heat transfer, however, in ad dition to working regularly on other subjects is a rather difficult task for which some years of meditation are necessary. The book must satisfy two requirements which are not easily reconciled. From the mathematical point of view, it must be written in accordance with standards of mathemati cal rigor and precision."

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

The articles in this volume focus on control theory of systems governed by nonlinear linear partial differential equations, identification and optimal design of such systems, and modelling of advanced materials. Optimal design of systems governed by PDEs is a relatively new area of study, now particularly relevant because of interest in optimization of fluid flow in domains of variable configuration, advanced and composite materials studies and "smart" materials which include possibilities for built in sensing and control actuation. The book will be of interest to both applied mathematicians and to engineers.

The basis for this book is a number of lectures given frequently by the author to third year students of the Department of Economics at Leningrad State University who specialize in economical cybernetics. The main purpose of this book is to provide the student with a relatively simple and easy-to-understand manual containing the basic mathematical machinery utilized in the theory of games. Practical examples (including those from the field of economics) serve mainly as an interpretation of the mathematical foundations of this theory rather than as indications of their actual or potential applicability. The present volume is significantly different from other books on the theory of games. The difference is both in the choice of mathematical problems as well as in the nature of the exposition. The realm of the problems is somewhat limited but the author has tried to achieve the greatest possible systematization in his exposition. Whenever possible the author has attempted to provide a game-theoretical argument with the necessary mathematical rigor and reasonable generality. Formal mathematical prerequisites for this book are quite modest. Only the elementary tools of linear algebra and mathematical analysis are used.

During the past decade the interaction between control theory and linear algebra has been ever increasing, giving rise to new results in both areas. As a natural outflow of this research, this book presents information on this interdisciplinary area. The cross-fertilization between control and linear algebra can be found in subfields such as Numerical Linear Algebra, Canonical Forms, Ring-theoretic Methods, Matrix Theory, and Robust Control. This book's editors were challenged to present the latest results in these areas and to find points of common interest. This volume reflects very nicely the interaction: the range of topics seems very wide indeed, but the basic problems and techniques are always closely connected. And the common denominator in all of this is, of course, linear algebra.
This book is suitable for both mathematicians and students.

After the IUTAM Symposium on Optimization in Structural Design held in Warsaw in 1973, it was clear to me that the time had come for organizing into a consistent body of thought the enormous quantity of results obtained in this domain, studied from so many different points of view, with so many different methods, and at so many levels of practical applicability. My colleague and friend Gianantonnio Sacchi from Milan and I met with Professor Prager in Savognin in July 1974, where I submitted to them my first ideas for a treatise on structural optimization: It should cover the whole domain from basic theory to practical applications, and deal with various materials, various types of structures, various functions required of the structures, and various types of cost . . Obviously, this was to be a team effort, to total three or four volumes, to be written in a balanced manner as textbooks and handbooks. Nothing similar existed at that time, and, indeed, nothing has been published to date. Professor Prager was immedi ately in favor of such a project. He agreed to write a first part on optimality criteria with me and to help me in the general organization of the series. Since Professor Sacchi was willing to write the text on variational methods, it remained to find authors for parts on the mathematical programming approach to structural optimization (and, more generally, on numerical methods) and on practical optimal design procedures in metal and concrete."

This IMA Volume in Mathematics and its Applications STOCHASTIC DIFFERENTIAL SYSTEMS, STOCHASTIC CONTROL THEORY AND APPLICATIONS is the proceedings of a workshop which was an integral part of the 1986-87 IMA program on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS. We are grateful to the Scientific Committee: Daniel Stroock (Chairman) WendeIl Flerning Theodore Harris Pierre-Louis Lions Steven Orey George Papanicolaou for planning and implementing an exciting and stimulating year-long program. We es pecially thank WendeIl Fleming and Pierre-Louis Lions for organizing an interesting and productive workshop in an area in which mathematics is beginning to make significant contributions to real-world problems. George R. Seil Hans Weinberger PREFACE This volume is the Proceedings of a Workshop on Stochastic Differential Systems, Stochastic Control Theory, and Applications held at IMA June 9-19,1986. The Workshop Program Commit tee consisted of W.H. Fleming and P.-L. Lions (co-chairmen), J. Baras, B. Hajek, J.M. Harrison, and H. Sussmann. The Workshop emphasized topics in the following four areas. (1) Mathematical theory of stochastic differential systems, stochastic control and nonlinear filtering for Markov diffusion processes. Connections with partial differential equations. (2) Applications of stochastic differential system theory, in engineering and management sci ence. Adaptive control of Markov processes. Advanced computational methods in stochas tic control and nonlinear filtering. (3) Stochastic scheduling, queueing networks, and related topics. Flow control, multiarm bandit problems, applications to problems of computer networks and scheduling of complex manufacturing operations."

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."

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

"The central fact is that we are planning agents." (M. Bratman, Intentions, Plans, and Practical Reasoning, 1987, p. 2) Recent arguments to the contrary notwithstanding, it seems to be the case that people-the best exemplars of general intelligence that we have to date do a lot of planning. It is therefore not surprising that modeling the planning process has always been a central part of the Artificial Intelligence enterprise. Reasonable behavior in complex environments requires the ability to consider what actions one should take, in order to achieve (some of) what one wants and that, in a nutshell, is what AI planning systems attempt to do. Indeed, the basic description of a plan generation algorithm has remained constant for nearly three decades: given a desciption of an initial state I, a goal state G, and a set of action types, find a sequence S of instantiated actions such that when S is executed instate I, G is guaranteed as a result. Working out the details of this class of algorithms, and making the elabora tions necessary for them to be effective in real environments, have proven to be bigger tasks than one might have imagined.

In this treatise we present the semigroup approach to quasilinear evolution equa of parabolic type that has been developed over the last ten years, approxi tions mately. It emphasizes the dynamic viewpoint and is sufficiently general and flexible to encompass a great variety of concrete systems of partial differential equations occurring in science, some of those being of rather 'nonstandard' type. In partic ular, to date it is the only general method that applies to noncoercive systems. Although we are interested in nonlinear problems, our method is based on the theory of linear holomorphic semigroups. This distinguishes it from the theory of nonlinear contraction semigroups whose basis is a nonlinear version of the Hille Yosida theorem: the Crandall-Liggett theorem. The latter theory is well-known and well-documented in the literature. Even though it is a powerful technique having found many applications, it is limited in its scope by the fact that, in concrete applications, it is closely tied to the maximum principle. Thus the theory of nonlinear contraction semigroups does not apply to systems, in general, since they do not allow for a maximum principle. For these reasons we do not include that theory."

Robust control originates with the need to cope with systems with modeling uncertainty. There have been several mathematical techniques developed for robust control system analysis. The articles in this volume cover all of the major research directions in the field.

Many devices (we say dynamical systems or simply systems) behave like black boxes: they receive an input, this input is transformed following some laws (usually a differential equation) and an output is observed. The problem is to regulate the input in order to control the output, that is for obtaining a desired output. Such a mechanism, where the input is modified according to the output measured, is called feedback. The study and design of such automatic processes is called control theory. As we will see, the term system embraces any device and control theory has a wide variety of applications in the real world. Control theory is an interdisci plinary domain at the junction of differential and difference equations, system theory and statistics. Moreover, the solution of a control problem involves many topics of numerical analysis and leads to many interesting computational problems: linear algebra (QR, SVD, projections, Schur complement, structured matrices, localization of eigenvalues, computation of the rank, Jordan normal form, Sylvester and other equations, systems of linear equations, regulariza tion, etc), root localization for polynomials, inversion of the Laplace transform, computation of the matrix exponential, approximation theory (orthogonal poly nomials, Pad6 approximation, continued fractions and linear fractional transfor mations), optimization, least squares, dynamic programming, etc. So, control theory is also a. good excuse for presenting various (sometimes unrelated) issues of numerical analysis and the procedures for their solution. This book is not a book on control."

Over the past several years, cooperative control and optimization has un questionably been established as one of the most important areas of research in the military sciences. Even so, cooperative control and optimization tran scends the military in its scope -having become quite relevant to a broad class of systems with many exciting, commercial, applications. One reason for all the excitement is that research has been so incredibly diverse -spanning many scientific and engineering disciplines. This latest volume in the Cooperative Systems book series clearly illustrates this trend towards diversity and creative thought. And no wonder, cooperative systems are among the hardest systems control science has endeavored to study, hence creative approaches to model ing, analysis, and synthesis are a must The definition of cooperation itself is a slippery issue. As you will see in this and previous volumes, cooperation has been cast into many different roles and therefore has assumed many diverse meanings. Perhaps the most we can say which unites these disparate concepts is that cooperation (1) requires more than one entity, (2) the entities must have some dynamic behavior that influences the decision space, (3) the entities share at least one common objective, and (4) entities are able to share information about themselves and their environment. Optimization and control have long been active fields of research in engi neering."

Proceedings of the IUTAM Symposium on Structural Optimization, Melbourne, Australia, February 9-13, 1988

This book is devoted to an investigation of control problems which can be described by ordinary differential equations and be expressed in terms of game theoretical notions. In these terms, a strategy is a control based on the feedback principle which will assure a definite equality for the controlled process which is subject to uncertain factors such as a move or a controlling action of the opponent. "Game" "Theoretical Control Problems" contains definitions and formalizations of differential games, existence for equilibrium and extensive discussions of optimal strategies. Formal definitions and statements are accompanied by suitable motivations and discussions of computational algorithms. The book is addessed to mathematicians, engineers, economists and other users of control theoretical and game theoretical notions.

Biotechnology has been labelled as one of the key technologies of the last two decades of the 20th Century, offering boundless solutions to problems ranging from food and agricultural production to pharmaceutical and medical applications, as well as environmental and bioremediation problems. Biological processes, however, are complex and the prevailing mechanisms are either unknown or poorly understood. This means that adequate techniques for data acquisition and analysis, leading to appropriate modeling and simulation packages that can be superimposed on the engineering principles, need to be routine tools for future biotechnologists. The present volume presents a masterly summary of the most recent work in the field, covering: instrumentation systems; enzyme technology; environmental biotechnology; food applications; and metabolic engineering.

This monograph is an introduction to optimal control theory for systems governed by vector ordinary differential equations. It is not intended as a state-of-the-art handbook for researchers. We have tried to keep two types of reader in mind: (1) mathematicians, graduate students, and advanced undergraduates in mathematics who want a concise introduction to a field which contains nontrivial interesting applications of mathematics (for example, weak convergence, convexity, and the theory of ordinary differential equations); (2) economists, applied scientists, and engineers who want to understand some of the mathematical foundations. of optimal control theory. In general, we have emphasized motivation and explanation, avoiding the "definition-axiom-theorem-proof" approach. We make use of a large number of examples, especially one simple canonical example which we carry through the entire book. In proving theorems, we often just prove the simplest case, then state the more general results which can be proved. Many of the more difficult topics are discussed in the "Notes" sections at the end of chapters and several major proofs are in the Appendices. We feel that a solid understanding of basic facts is best attained by at first avoiding excessive generality. We have not tried to give an exhaustive list of references, preferring to refer the reader to existing books or papers with extensive bibliographies. References are given by author's name and the year of publication, e.g., Waltman [1974].

The articles in this volume cover recent work in the area of flow control from the point of view of both engineers and mathematicians. These writings are especially timely, as they coincide with the emergence of the role of mathematics and systematic engineering analysis in flow control and optimization. Recently this role has significantly expanded to the point where now sophisticated mathematical and computational tools are being increasingly applied to the control and optimization of fluid flows. These articles document some important work that has gone on to influence the practical, everyday design of flows; moreover, they represent the state of the art in the formulation, analysis, and computation of flow control problems. This volume will be of interest to both applied mathematicians and to engineers.