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11/24/2010 -
Stabilization, Optimal and Robust Control: Theory and Applications in Biological and Physical Sciences (2008)
Aziz Belmiloudi, author
ISBN 9781848003439. Springer-Verlag, London. 2008. Hardcover. 502 pages.
The topic of this book is the mathematical modeling of systems and processes for which the dynamics are describable by non-linear partial differential equations. Today, the modeling of many materials systems and processes of interest requires models that can account for multiple scales of length and time, non-linear dynamics and kinetics, time delays and hysteresis effects, and uncertainties of parameters. This book deals with the basic principles of the development and analysis of such models within the framework of “robust control theory”, an outgrowth of classical optimal control techniques of applied mathematics, the goal of which is to estimate performance changes of a dynamical system with changing parameters and functions and to develop alternatives that are insensitive to changes in the system in order to maintain stability and performance.
There many such systems and processes of importance to materials scientists and engineers, and this book and several are considered explicitly in this book: vortex dynamics in superconductors, multiscale modeling of alloy solidification, and semiconductor melt flow in crystal growth. Several more general systems of interest to materials scientists, biologists, and chemists are also considered explicitly. One such is Lotka-Volterra systems of equations describing population dynamics; these are valuable for describing processes of radiation damage and annealing, ion-implantation, reaction-diffusion processes and other materials processes and phenomena. There is also a discussion of heat transfer and temperature distributions in biological tissues that is easily adaptable to similar materials problems.
The book is intended to be useful to “researchers in mathematics, physics, biology, and chemistry, and to professionals involved in complex problems in fluid mechanics, biological systems, and material sciences.” The book is constructed so as to combine the general theory of control, optimization theory, the modeling process, and the theory of time-dependent coupled partial differential equations into a “complete unified theory.” It is intended to be accessible to the non-specialist. The book is divided into three more-or-less independent parts: essential mathematical foundations, elements of classical optimum control theory and robust control theory for evolutive systems, and applications to biological and physical systems. The last section is the most interesting for materials scientists and engineers, and contains explicit analyses and illustrations of: 1) time-dependent Ginzburg-Landau theory of superconductors in magnetic fields and the effects of perturbations, fluctuations, and noise on the flow of vortices under the control of magnetic fields and electric currents; 2) multi-scale modeling of the effects of thermal fluctuations and impurities on alloy solidification, with the explicit formulation and analysis of a 2-dimensional, non-linear, time-dependent, phase-field model of the Warren-Boettinger type; 3)heat transfer and temperature distributions in biological tissues; and 4) the formulation of optimal solutions for Lotka-Volterra popultations dynamics; and 5) conditions for the stabilization of melt-flow against perturbations and temperature fluctuations during semiconductor crystal growth.
The approach taken in the book is heavily applied-mathematical, involving such matters as existence and uniqueness theorems for solutions of models, stability conditions, and their consideration in the construction of models, and so forth, with a minimal amount of physical considerations, as is understandable in view of the author’s places of emplyment in French mathematical research institutions: Institut de Recherche Mathematique de Rennes and the Centres de Mathemaiques Insitut National des Sciences Appliquees de Rennes. The approach taken is also primarily foundational and analytical in nature, with only one small section of remarks devoted to numerical techniques. The book is packed full of equations, but contains only a single diagram, a schematic illustration of the relationship between tumor vascularity and blood flow direction, in a section devoted to thermal damage calculations in thermotherapy. The book does contain a useful selection of references which can fill out the physical understanding of the models discussed, as well as their implementation. The production qualities of the book are excellent, as is usual with books published by Springer.
In my opinion, this is a useful book for seasoned theorists and modelers of materials and materials processes. It is definitely not a book for the novice modeler in these areas, unless it is an applied mathematician who needs an introduction to these methods of robust control and optimization without much discussion of the physical bases of the problems. However, the theoretical materials physicist will need a reasonably high level of mathematical sophistication to benefit from this book, but for the experienced theorist or modeler in materials science and engineering, it will be a valuable and interesting resource for the mathematical foundations and new methods for developing new models and analyzing old ones.
For more on Stabilization, Optimal and Robust Control: Theory and Applications in Biological and Physical Sciences, visit the Springer-Verlag web site. |
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