An Introduction To Infinite Dimensional Linear Systems Theory

Infinite dimensional systems is now an established area of research. Given the recent trend in systems theory and in applications towards a synthesis of time- and frequency-domain methods, there is a need for an introductory text which treats both state-space and frequency-domain aspects in an integrated fashion. The authors' primary aim is to write an introductory textbook for a course on infinite dimensional linear systems. An important consideration by the authors is that their book should be accessible to graduate engineers and mathematicians with a minimal background in functional analysis. Consequently, all the mathematical background is summarized in an extensive appendix. For the majority of students, this would be their only acquaintance with infinite dimensional systems.

This book provides a self-contained introduction to the theory of infinite-dimensional systems theory and its applications to port-Hamiltonian systems. The textbook starts with elementary known results, then progresses smoothly to advanced topics in current research. Many physical systems can be formulated using a Hamiltonian framework, leading to models described by ordinary or partial differential equations. For the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take into account this interaction with the environment. This book is the first textbook on infinite-dimensional port-Hamiltonian systems. An abstract functional analytical approach is combined with the physical approach to Hamiltonian systems. This combined approach leads to easily verifiable conditions for well-posedness and stability. The book is accessible to graduate engineers and mathematicians with a minimal background in functional analysis. Moreover, the theory is illustrated by many worked-out examples.

This book deals with the stabilization issue of infinite dimensional dynamical systems both at the theoretical and applications levels. Systems theory is a branch of applied mathematics, which is interdisciplinary and develops activities in fundamental research which are at the frontier of mathematics, automation and engineering sciences. It is everywhere, innumerable and daily, and moreover is there something which is not system: it is present in medicine, commerce, economy, psychology, biological sciences, finance, architecture (construction of towers, bridges, etc.), weather forecast, robotics, automobile, aeronautics, localization systems and so on. These are the few fields of application that are useful and even essential to our society. It is a question of studying the behavior of systems and acting on their evolution. Among the most important notions in system theory, which has attracted the most attention, is stability. The existing literature on systems stability is quite important, but disparate, and the purpose of this book is to bring together in one document the essential results on the stability of infinite dimensional dynamical systems. In addition, as such systems evolve in time and space, explorations and research on their stability have been mainly focused on the whole domain in which the system evolved. The authors have strongly felt that, in this sense, important considerations are missing: those which consist in considering that the system of interest may be unstable on the whole domain, but stable in a certain region of the whole domain. This is the case in many applications ranging from engineering sciences to living science. For this reason, the authors have dedicated this book to extension of classical results on stability to the regional case. This book considers a very important issue, which is that it should be accessible to mathematicians and to graduate engineering with a minimal background in functional analysis. Moreover, for the majority of the students, this would be their only acquaintance with infinite dimensional system. Accordingly, it is organized by following increasing difficulty order. The two first chapters deal with stability and stabilization of infinite dimensional linear systems described by partial differential equations. The following chapters concern original and innovative aspects of stability and stabilization of certain classes of systems motivated by real applications, that is to say bilinear and semi-linear systems. The stability of these systems has been considered from a global and regional point of view. A particular aspect concerning the stability of the gradient has also been considered for various classes of systems. This book is aimed at students of doctoral and master’s degrees, engineering students and researchers interested in the stability of infinite dimensional dynamical systems, in various aspects.

Infinite-dimensional systems is a well established area of research with an ever increasing number of applications. Given this trend, there is a need for an introductory text treating system and control theory for this class of systems in detail. This textbook is suitable for courses focusing on the various aspects of infinite-dimensional state space theory. This book is made accessible for mathematicians and post-graduate engineers with a minimal background in infinite-dimensional system theory. To this end, all the system theoretic concepts introduced throughout the text are illustrated by the same types of examples, namely, diffusion equations, wave and beam equations, delay equations and the new class of platoon-type systems. Other commonly met distributed and delay systems can be found in the exercise sections. Every chapter ends with such a section, containing about 30 exercises testing the theoretical concepts as well. An extensive account of the mathematical background assumed is contained in the appendix.

Based on well-known lectures given at Scuola Normale Superiore in Pisa, this book introduces analysis in a separable Hilbert space of infinite dimension. It starts from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate basic stochastic dynamical systems and Markov semi-groups, paying attention to their long-time behavior.

This book treats the theory of global attractors, a recent development in the theory of partial differential equations, in a way that also includes much of the traditional elements of the subject. As such it gives a quick but directed introduction to some fundamental concepts, and by the end proceeds to current research problems. Since the subject is relatively new, this is the first book to attempt to treat these various topics in a unified and didactic way. It is intended to be suitable for first year graduate students.