Weijiu
Liu (ÁõÍþ¾Å)
Department
of Mathematics
University of Central Arkansas
201 Donaghey Avenue
Conway, AR 72035, USA
TEL (501) 450-5684
FAX (501) 450-5662
E-Mail: weijiul@uca.edu
o
Feedback
control of the blood glucose system
o
Feedback
control of the intracellular calcium system in yeast cells
o
Feedback
control of the intracellular calcium system in non-excitable cells
o
Feedback
control of the intracellular sterol system in yeast cells
Preface
Mathematical
control theory of applied partial differential equations is built on linear and
nonlinear functional analysis and many existence theorems in control theory
result from applications of theorems in functional analysis. This makes control
theory inaccessible to students who do not have a background in functional
analysis.
Many
advanced control theory books on infinite-dimensional systems were written, using
functional analysis and semigroup theory, and control theory was presented in
an abstract setting. This motivates me to write this text for control theory
classes in the way to present control theory by concrete examples and try to
minimize the use of functional analysis. Functional analysis is not assumed and
any analysis included here is elementary, using calculus such as integration by
parts. The material presented in this text is just a simplification of the
material from the existing advanced control books. Thus this text is accessible
to senior undergraduate students and first-year graduate students in applied
mathematics, who have taken linear algebra and ordinary and partial
differential equations.
Elementary
functional analysis is presented in Chapter 2. This material is required to
present the control theory of partial differential equations. Since many
control concepts and theories for partial differential equations are
transplanted from finite-dimensional control systems, a brief introduction to
feedback control of these systems is presented in Chapter 3. The topics covered
in this chapter include controllability, observability, stabilizability, pole
placement, and quadratic optimal control. Theories about the feedback
stabilization of linear reaction-convection-diffusion equations are presented
in Chapter 4. Both interior and boundary control problems are addressed. The
methods employed to handle the problems include eigenfunction expansions,
integral transforms, and optimal control. Finally, theories about feedback
stabilization of linear wave equations are presented in Chapters 5 and 6.
First, the one-dimensional wave equations are considered, then higher
dimensional wave equations follows, since it is easier to use the
one-dimensional equations to illustrate the theories and methods. The perturbed
energy method is emphasized to deal with both interior and boundary control
problems while the optimal control technique is used.
This
text can be used as a textbook for an introductory one-semester graduate course
on control theory of partial differential equations. If students do not have a
background in finite-dimensional control systems, one can cover Sections
3.1-3.7 of Chapter 3, Sections 4.1-4.3 of Chapter 4, all sections of Chapter 5,
and Sections 6.1-6.3 of Chapter 6. Material from Chapter 2 can be introduced
whenever needed, and all material about optimal control can be skipped. If
students have had a background in both functional analysis and
finite-dimensional control systems, one can cover all of Chapters 4, 5, 6 and
give a brief review of Chapters 2 and 3.
Inevitably,
the material is selected from the topics I have worked on or am most familiar
with, and many other important topics are not covered. Control theory for other
applied partial differential equations, such as elastic equations,
thermoelastic equations, viscoelastic equations, Schr¨odinger’s equations, and
Navier-Stokes equations, is not included. In addition, exact and approximate
controllability is not discussed since it is difficult to present it in an
elementary way.
I
thank my PhD advisor, Dr. Graham H. Williams, for introducing me to the area of
control theory of partial differential equations. I thank Drs. George Haller,
Mirsolav Krstic, and Enrique Zuazua for having directed my research and
providing stimulating feedback on my work. I thank two reviewers for their
hard, serious work of reading and evaluating this text and giving constructive
suggestions. I thank my students, such as Jingvoon Chen, for using this text and
correcting mistakes. I thank Ms. Peggy Arrigo and Dr. Danny Arrigo for their
language-editing. Finally, I thank my family for their constant support.
In general terms, control theory
is a separate branch of mathematics and can be described as the study of how to
design the process of influencing the behavior of a physical system to achieve
a desired goal. Control theory consists of three main components: feedback
control, controllability, and optimal control. My research interest
focuses on feedback control because, in comparison with controllability and
optimal control, it is mathematically simpler and physically more applicable in
solving control problems in engineering and sciences.
Feedback control refers to an operation that, in the presence of
disturbance, tends to reduce the difference between the output of a system and
some reference output and does so on the basis of this difference.
Feedback control theory has developed an extensive literature covering its
various concepts, theories, and applications. Significant works in the early
stage of feedback control were due to Watt, Minorsky, Hazen, and Nyquist, among
many others. The first significant work in automatic control was James Watt's
centrifugal governor for the speed control of a steam engine in the eighteenth
century.
In the 1930s, automatic control was still an engineer's dominion. Feedback
controllers were designed and used by process engineers to regulate
temperatures and pressures, by mechanical engineers to control the speed of
engines, and by telephone engineers to build linear amplifiers. Despite their
wide use, a large fraction of feedback control design consisted of ``trial and
error" methods with little analysis involved if any at all. In 1934,
Ivanoff stated that ``the science of the automatic regulation of
temperature is at present in the anomalous position of having erected a vast
practical edifice on negligible theoretical foundations."
While feedback controllers were generally specialized mechanisms designed to
solve immediate industrial problems, there were also a small number of
researchers examining the theory behind these controllers. In the 1920s, while
studying the automatic steering of ships at sea, Nicolas Minorsky used
differential equations to model the dynamics of the ship and
proportional-integral-derivative (PID) controllers and then investigated their
stability. In the 1930s, using open-loop responses to steady-state sinusoidal
inputs, Harry Nyquist developed a theory about the stability of closed-loop
systems. Harold Hazen introduced the term servomechanisms for position control
systems with a changing input and designed relay servomechanisms to track the
changing input. During the decade of the 1940s, frequency-response methods
represented by the Bode diagram method due to Hendrick Bode were developed.
From the end of the 1940s to the early 1950s, the root-locus method due to
Walter R. Evans was established. The frequency-response and root-locus methods
made it possible for engineers to design linear single-input-single-output
closed-loop control systems that satisfy a set of performance requirements.
These early development of control theory constitutes classical control theory
with the frequency-response and root-locus
methods as its core.
Modern feedback control systems have becomes more and more complex. These
systems have many inputs and outputs. The stringent requirements on accuracy
lead to distributed parameter systems in which partial differential equations
is required to model the physical systems. Therefore classical control theory,
which deals only with single-input-single-output control systems, becomes
powerless for the modern complex control systems. Since about 1960, modern
control theory has been developed to cope with the increasing complexity of
multiple-inputs-multiple-outputs control systems. Unlike classical control theory
that is based on frequency-domain analysis, modern control theory is based on
time-domain analysis and synthesis using state variables.
Since the 1960s, control theory of distributed parameter systems has followed
right on the heels of that of finite dimensional dynamical systems, but with
slower and heavier tread. Many control concepts and theories in finite
dimensional systems have been transplanted to distributed parameter systems.
Control theory for many distributed parameter systems, such as the wave
equation, reaction-convection-diffusion equations, elastic equations,
thermoelastic equations, viscoelastic equations, Schrödinger's equations,
and Navier-Stokes equations, has been well developed.
Recently it has become appreciated that control theory can be used to analyze
non-engineering systems, such as biological systems and control theory has been
inspired by these biological systems.