Showing papers by "Anthony N. Michel published in 2011"

Relaxation of Hypotheses in LaSalle–Krasovskii-Type Invariance Results

Abstract: The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets E with increasing time. This allows that between the time instants determined by E, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to ...

Algebra and Analysis for Engineers and Scientists

Abstract: "This is an intermediate level text, with exercises, whose avowed purpose is to provide the science and engineering graduate student with an appropriate modern mathematical (analysis and algebra) background in a succinct, but nontrivial, manner.... [T]he book is quite thorough and can serve as a text, for self-study, or as a reference." Mathematical Reviews Written for graduate and advanced undergraduate students in engineering and science, this classic book focuses primarily on set theory, algebra, and analysis. Useful as a course textbook, for self-study, or as a reference, the work is intended to: * provide readers with appropriate mathematical background for graduate study in engineering or science; * allow students in engineering or science to become familiar with a great deal of pertinent mathematics in a rapid and efficient manner without sacrificing rigor; * give readers a unified overview of applicable mathematics, enabling them to choose additional, advanced topical courses in mathematics more intelligently. Whereas these objectives for writing this book were certainly pertinent over twenty years ago when the work was first published, they are even more compelling now. Todays graduate students in engineering or science are expected to be more knowledgeable and sophisticated in mathematics than students in the past. Moreover, todays graduate students in engineering or science are expected to be familiar with a great deal of ancillary material (primarily in the computer science area), acquired in courses that did not even exist a couple of decades ago. The book is divided into three parts: set theory (Chapter 1), algebra (Chapters 24), and analysis (Chapters 57). The first two chapters deal with the fundamental concepts of sets, functions, relations and equivalence relations, and algebraic structures. Chapters 3 and 4 cover vector spaces and linear transformations, and finite-dimensional vector spaces and matrices. The last three chapters investigate metric spaces, normed and inner product spaces, and linear operators. Because of its flexible structure, Algebra and Analysis for Engineers and Scientists may be used either in a one- or two-semester course by deleting appropriate sections, taking into account the students backgrounds and interests. A generous number of exercises have been integrated into the text, and a section of references and notes is provided at the end of each chapter. Applications of algebra and analysis having a broad appeal are also featured, including topics dealing with ordinary differential equations, integral equations, applications of the contraction mapping principle, minimization of functionals, an example from optimal control, and estimation of random variables. Supplementary material for students and instructors is available at http://Michel.Herget.net.