The aim of Stability of Finite and Infinite Dimensional Systems is to provide new tools for specialists in control system theory, stability theory of ordinary and partial differential equations, and differential-delay equations. Stability of Finite and Infinite Dimensional Systems is the first book that gives a systematic exposition of the approach to stability analysis which is based on estimates for matrix-valued and operator-valued functions, allowing us to investigate various classes of finite and infinite dimensional systems from the unified viewpoint. This book contains solutions to the problems connected with the Aizerman and generalized Aizerman conjectures and presents fundamental results by A. Yu. Levin for the stability of nonautonomous systems having variable real characteristic roots. Stability of Finite and Infinite Dimensional Systems is intended not only for specialists in stability theory, but for anyone interested in various applications who has had at least a first-year graduate-level course in analysis.

Dissipativity, as a natural mechanism of energy interchange is common to many physical systems that form the basis of modern automated control applications. Over the last decades it has turned out as a useful concept that can be generalized and applied in an abstracted form to very different system setups, including ordinary and partial differential equation models. In this monograph, the basic notions of stability, dissipativity and systems theory are connected in order to establish a common basis for designing system monitoring and control schemes. The approach is illustrated with a set of application examples covering finite and infinite-dimensional models, including a ship steering model, the inverted pendulum, chemical and biological reactors, relaxation oscillators, unstable heat equations and first-order hyperbolic integro-differential equations.

This book reports on recent achievements in stability and feedback stabilization of infinite systems. In particular emphasis is placed on second order partial differential equations, such as Euler-Bernoulli beam equations, which arise from vibration control of flexible robots arms and large space structures. Various control methods such as sensor feedback control and dynamic boundary control are applied to stabilize the equations. Many new theorems and methods are included in the book. Proof procedures of existing theorems are simplified, and detailed proofs have been given to most theorems. New results on semigroups and their stability are presented, and readers can learn several useful techniques for solving practical engineering problems. Until now, the recently obtained research results included in this book were unavailable in one volume. This self-contained book is an invaluable source of information for all those who are familiar with some basic theorems of functional analysis.

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.

Stochastic differential equations in infinite dimensional spaces are motivated by the theory and analysis of stochastic processes and by applications such as stochastic control, population biology, and turbulence, where the analysis and control of such systems involves investigating their stability. While the theory of such equations is well establ

The quadratic cost optimal control problem for systems described by linear ordinary differential equations occupies a central role in the study of control systems both from the theoretical and design points of view. The study of this problem over an infinite time horizon shows the beautiful interplay between optimality and the qualitative properties of systems such as controllability, observability and stability. This theory is far more difficult for infinite-dimensional systems such as systems with time delay and distributed parameter systems. In the first place, the difficulty stems from the essential unboundedness of the system operator. Secondly, when control and observation are exercised through the boundary of the domain, the operator representing the sensor and actuator are also often unbounded. The present book, in two volumes, is in some sense a self-contained account of this theory of quadratic cost optimal control for a large class of infinite-dimensional systems. Volume I deals with the theory of time evolution of controlled infinite-dimensional systems. It contains a reasonably complete account of the necessary semigroup theory and the theory of delay-differential and partial differential equations. Volume II deals with the optimal control of such systems when performance is measured via a quadratic cost. It covers recent work on the boundary control of hyperbolic systems and exact controllability. Some of the material covered here appears for the first time in book form. The book should be useful for mathematicians and theoretical engineers interested in the field of control.

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.