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This volume gathers contributions in the field of partial
differential equations, with a focus on mathematical models in
phase transitions, complex fluids and thermomechanics. These
contributions are dedicated to Professor Gianni Gilardi on the
occasion of his 70th birthday. It particularly develops the
following thematic areas: nonlinear dynamic and stationary
equations; well-posedness of initial and boundary value problems
for systems of PDEs; regularity properties for the solutions;
optimal control problems and optimality conditions; feedback
stabilization and stability results. Most of the articles are
presented in a self-contained manner, and describe new achievements
and/or the state of the art in their line of research, providing
interested readers with an overview of recent advances and future
research directions in PDEs.
This volume gathers contributions in the field of partial
differential equations, with a focus on mathematical models in
phase transitions, complex fluids and thermomechanics. These
contributions are dedicated to Professor Gianni Gilardi on the
occasion of his 70th birthday. It particularly develops the
following thematic areas: nonlinear dynamic and stationary
equations; well-posedness of initial and boundary value problems
for systems of PDEs; regularity properties for the solutions;
optimal control problems and optimality conditions; feedback
stabilization and stability results. Most of the articles are
presented in a self-contained manner, and describe new achievements
and/or the state of the art in their line of research, providing
interested readers with an overview of recent advances and future
research directions in PDEs.
Hysteresis is an exciting and mathematically challenging phenomenon
that oc curs in rather different situations: jt, can be a byproduct
offundamental physical mechanisms (such as phase transitions) or
the consequence of a degradation or imperfection (like the play in
a mechanical system), or it is built deliberately into a system in
order to monitor its behaviour, as in the case of the heat control
via thermostats. The delicate interplay between memory effects and
the occurrence of hys teresis loops has the effect that hysteresis
is a genuinely nonlinear phenomenon which is usually non-smooth and
thus not easy to treat mathematically. Hence it was only in the
early seventies that the group of Russian scientists around M. A.
Krasnoselskii initiated a systematic mathematical investigation of
the phenomenon of hysteresis which culminated in the fundamental
monograph Krasnoselskii-Pokrovskii (1983). In the meantime, many
mathematicians have contributed to the mathematical theory, and the
important monographs of 1. Mayergoyz (1991) and A. Visintin (1994a)
have appeared. We came into contact with the notion of hysteresis
around the year 1980."
The present monograph is intended to provide a comprehensive and
accessible introduction to the optimization of elliptic systems.
This area of mathematical research, which has many important
applications in science and technology. has experienced an
impressive development during the past two decades. There are
already many good textbooks dealing with various aspects of optimal
design problems. In this regard, we refer to the works of Pironneau
[1984], Haslinger and Neittaanmaki [1988], [1996], Sokolowski and
Zolksio [1992], Litvinov [2000], Allaire [2001], Mohammadi and
Pironneau [2001], Delfour and Zolksio [2001], and Makinen and
Haslinger [2003]. Already Lions [I9681 devoted a major part of his
classical monograph on the optimal control of partial differential
equations to the optimization of elliptic systems. Let us also
mention that even the very first known problem of the calculus of
variations, the brachistochrone studied by Bernoulli back in 1696.
is in fact a shape optimization problem. The natural richness of
this mathematical research subject, as well as the extremely large
field of possible applications, has created the unusual situation
that although many important results and methods have already been
est- lished, there are still pressing unsolved questions. In this
monograph, we aim to address some of these open problems; as a
consequence, there is only a minor overlap with the textbooks
already existing in the field.
The international Conference on Optimal Control of Coupled Systems
of Partial Di?erential Equations was held at the Mathematisches
Forschungsinstitut Ob- wolfach (www.mfo.de) from April, 17 to 23,
2005. The scienti?c program included 30 talks coveringvarious
topics as controllability,feedback-control,optimality s- tems,
model-reduction techniques, analysis and optimal control of ?ow
problems and ?uid-structure interactions, as well as problems of
shape and topology op- mization. The applications discussed during
the conference range from the op- mization and control of quantum
mechanical systems, the design of piezo-electric acoustic
micro-mechanical devices, optimal control of crystal growth, the
control of bodies immersed into a ?uid to airfoil design and much
more. Thus the app- cations are across all time and length scales.
Optimization and control of systems governed by partial di?erential
eq- tions and more recently by variational inequalities is a very
active ?eld of research in Applied Mathematics, in particular in
numerical analysis, scienti?c comp- ing and optimization. In order
to able to handle real-world applications, scalable and
parallelizable algorithms have to be designed, implemented and
validated. This requires an in-depth understanding of both the
theoretical properties and the numerical realization of such
structural insights. Therefore, a 'core' devel- ment within the
?eld of optimization with PDE-constraints such as the analysis of
control-and-state-constrained problems, the role of obstacles,
multi-phases etc. and an interdisciplinary 'diagonal' bridging
regarding applications and numerical simulation are most important.
The present monograph is intended to provide a comprehensive and
accessible introduction to the optimization of elliptic systems.
This area of mathematical research, which has many important
applications in science and technology. has experienced an
impressive development during the past two decades. There are
already many good textbooks dealing with various aspects of optimal
design problems. In this regard, we refer to the works of Pironneau
[1984], Haslinger and Neittaanmaki [1988], [1996], Sokolowski and
Zolksio [1992], Litvinov [2000], Allaire [2001], Mohammadi and
Pironneau [2001], Delfour and Zolksio [2001], and Makinen and
Haslinger [2003]. Already Lions [I9681 devoted a major part of his
classical monograph on the optimal control of partial differential
equations to the optimization of elliptic systems. Let us also
mention that even the very first known problem of the calculus of
variations, the brachistochrone studied by Bernoulli back in 1696.
is in fact a shape optimization problem. The natural richness of
this mathematical research subject, as well as the extremely large
field of possible applications, has created the unusual situation
that although many important results and methods have already been
est- lished, there are still pressing unsolved questions. In this
monograph, we aim to address some of these open problems; as a
consequence, there is only a minor overlap with the textbooks
already existing in the field.
This monograph contributes to the mathematical analysis of systems
exhibiting hysteresis effects and phase transitions. Its main part
begins with a detailed study of models for scalar rate independent
hysteresis in the form of hysteresis operators. Applications to
ferromagnetism, elastoplasticity and fatigue analysis are
presented, and two representative distributed systems with
hysteresis operator are discussed. The attention then shifts to the
mechanisms of energy dissipation and transformation that induce a
hysteretic behavior in continuous media undergoing phase
transitions. After an introduction to phenomenological
thermodynamic theories of phase transitions, in particular, the
Landau-Ginzburg theory and phase field models, several specific
models are discussed in detail. These include Falk's model for the
hysteresis in shape memory alloys and the phase field models due to
Caginalp and Penrose-Fife. The latter are studied both for
conserved and non-conserved order parameters. A chapter presenting
a mathematical model for the austenite-pearlite and
austenite-martensite phase transitions in eutectoid carbon steels
concludes the book.
Interest in the area of control of systems defined by partial
differential Equations has increased strongly in recent years. A
major reason has been the requirement of these systems for sensible
continuum mechanical modelling and optimization or control
techniques which account for typical physical phenomena. Particular
examples of problems on which substantial progress has been made
are the control and stabilization of mechatronic structures, the
control of growth of thin films and crystals, the control of Laser
and semi-conductor devices, and shape optimization problems for
turbomachine blades, shells, smart materials and microdiffractive
optics. This volume contains original articles by world reknowned
experts in the fields of optimal control of partial differential
equations, shape optimization, numerical methods for partial
differential equations and fluid dynamics, all of whom have
contributed to the analysis and solution of many of the problems
discussed. The collection provides a state-of-the-art overview of
the most challenging and exciting recent developments in the field.
It is geared towards postgraduate students and researchers dealing
with the theoretical and practical aspects of a wide variety of
high technology problems in applied mathematics, fluid control,
optimal design, and computer modelling.
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