Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, Hybrid Dynamical Systems unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

Hybrid Dynamical Systems: Modeling, Stability, and Robustness

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

Commuting Mappings and Fixed Points

This work deals with the topic of information processing over graphs. The presentation is largely self-contained and covers results that relate to the analysis and design of multi-agent networks for the distributed solution of optimization, adaptation, and learning problems from streaming data through localized interactions among agents. The results derived in this work are useful in comparing network topologies against each other, and in comparing networked solutions against centralized or batch implementations. There are many good reasons for the peaked interest in distributed implementations, especially in this day and age when the word "network" has become commonplace whether one is referring to social networks, power networks, transportation networks, biological networks, or other types of networks. Some of these reasons have to do with the benefits of cooperation in terms of improved performance and improved resilience to failure. Other reasons deal with privacy and secrecy considerations where agents may not be comfortable sharing their data with remote fusion centers. In other situations, the data may already be available in dispersed locations, as happens with cloud computing. One may also be interested in learning through data mining from big data sets. Motivated by these considerations, this work examines the limits of performance of distributed solutions and discusses procedures that help bring forth their potential more fully. The presentation adopts a useful statistical framework and derives performance results that elucidate the mean-square stability, convergence, and steady-state behavior of the learning networks. At the same time, the work illustrates how distributed processing over graphs gives rise to some revealing phenomena due to the coupling effect among the agents. These phenomena are discussed in the context of adaptive networks, along with examples from a variety of areas including distributed sensing, intrusion detection, distributed estimation, online adaptation, network system theory, and machine learning.

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Adaptation, Learning, and Optimization Over Networks

Multidimensional incentive compatibility and mechanism design

This is a lucid account of the highlights in the historical development of the calculus from ancient to modern times from the beginnings of geometry in antiquity to the nonstandard analysis of the twentieth century. It emphasizes the genesis and evolution of both fundamental concepts and computational techniques. The intended audience includes not only students of the history of mathematics, but also the wider mathematical community, specifically those who study, teach and use calculus. Among the distinctive features of this exposition are historically motivated exercises and carefully chosen illustrative examples. Numerous sections of the book are suitable for use in courses in introductory and advanced calculus as well as the general history of mathematics.

The Historical Development of the Calculus

This course could be called Advanced Calculus for the Real World, which has at least three dimensions. The prerequisites are Math 4031 (Advanced Calculus I) and Math 2085 (Linear Algebra), or their equivalents. The topics will include properties of Euclidean space, continuous functions, differential calculus in n-dimensional space, the inverse and implicit function theorems, and the Riemann integral in n-dimensions if time permits. This is a theorem-proving course, like Math 4031: Homework will be collected, corrected carefully and returned. The text will be my own set of lecture notes and exercises, which will be available locally at low cost. Math majors with the Mathematics Concentration can complete the advanced calculus component of the upper division requirements with 4031 followed by either 4035 or 4032. The Department recommends but does not require all three courses for students intending to pursue graduate study in mathematics.

Advanced calculus of several variables

Preface 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Particular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substitution Methods and Exact Equations 1.7 Population Models 1.8 Acceleration-Velocity Models 2 Linear Equations of Higher Order 2.1 Introduction: Second-Order Linear Equations 2.2 General Solutions of Linear Equations 2.3 Homogeneous Equations with Constant Coefficients 2.4 Mechanical Vibrations 2.5 Nonhomogeneous Equations and Undetermined Coefficients 2.6 Forced Oscillations and Resonance 2.7 Electrical Circuits 2.8 Endpoint Problems and Eigenvalues 3 Power Series Methods 3.1 Introduction and Review of Power Series 3.2 Series Solutions Near Ordinary Points 3.3 Regular Singular Points 3.4 Method of Frobenius: The Exceptional Cases 3.5 Bessel's Equation 3.6 Applications of Bessel Functions 4 LaplaceTransform Methods 4.1 Laplace Transforms and Inverse Transforms 4.2 Transformation of Initial Value Problems 4.3 Translation and Partial Fractions 4.4 Derivatives, Integrals, and Products of Transforms 4.5 Periodic and Piecewise Continuous Input Functions 4.6 Impulses and Delta Functions 5 Linear Systems of Differential Equations 5.1 First-Order Systems and Applications 5.2 The Method of Elimination 5.3 Matrices and Linear Systems 5.4 The Eigenvalue Method for Homogeneous Systems 5.5 Second-Order Systems and Mechanical Applications 5.6 Multiple Eigenvalue Solutions 5.7 Matrix Exponentials and Linear Systems 5.8 Nonhomogeneous Linear Systems 6 Numerical Methods 6.1 Numerical Approximation: Euler's Method 6.2 A Closer Look at the Euler Method 6.3 The Runge-Kutta Method 6.4 Numerical Methods for Systems 7 Nonlinear Systems and Phenomena 7.1 Equilibrium Solutions and Stability 7.2 Stability and the Phase Plane 7.3 Linear and Almost Linear Systems 7.4 Ecological Models: Predators and Competitors 7.5 Nonlinear Mechanical Systems 7.6 Chaos in Dynamical Systems 8 Fourier Series Methods 8.1 Periodic Functions and Trigonometric Series 8.2 General Fourier Series and Convergence 8.3 Fourier Sine and Cosine Series 8.4 Applications of Fourier Series 8.5 Heat Conduction and Separation of Variables 8.6 Vibrating Strings and the One-Dimensional Wave Equation 8.7 Steady-State Temperature and Laplace's Equation 9 Eigenvalues and Boundary Value Problems 9.1 Sturm-Liouville Problems and Eigenfunction Expansions 9.2 Applications of Eigenfunction Series 9.3 Steady Periodic Solutions and Natural Frequencies 9.4 Cylindrical Coordinate Problems 9.5 Higher-Dimensional Phenomena References for Further Study Appendix: Existence and Uniqueness of Solutions Answers to Selected Problems Index I-1

Elementary Differential Equations with Boundary Value Problems

Elementary differential equations with applications

1. First-Order Differential Equations. Differential Equations and Mathematical Model. Integrals as General and Particular Solutions. Direction Fields and Solution Curves. Separable Equations and Applications. Linear First-Order Equations. Substitution Methods and Exact Equations. 2. Mathematical Models and Numerical Methods. Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method. The Runge-Kutta Method. 3. Linear Systems and Matrices. Introduction to Linear Systems. Matrices and Gaussian Elimination. Reduced Row-Echelon Matrices. Matrix Operations. Inverses of Matrices. Determinants. Linear Equations and Curve Fitting. 4. Vector Spaces. The Vector Space R 3. The Vector Space Rn and Subspaces. Linear Combinations and Independence of Vectors. Bases and Dimension for Vector Spaces. General Vector Spaces. 5. Linear Equations of Higher Order. Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Undetermined Coefficients and Variation of Parameters. Forced Oscillations and Resonance. 6. Eigenvalues and Eigenvectors. Introduction to Eigenvalues. Diagonalization of Matrices. Applications Involving Powers of Matrices. 7. Linear Systems of Differential Equations. First-Order Systems and Applications. Matrices and Linear Systems. The Eigenvalue Method for Linear Systems. Second-Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Numerical Methods for Systems. 8. Matrix Exponential Methods. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems. Spectral Decomposition Methods. 9. Nonlinear Systems and Phenomena. Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. 10. Laplace Transform Methods. Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. 11. Power Series Methods. Introduction and Review of Power Series. Power Series Solutions. Frobenius Series Solutions. Bessel's Equation. References for Further Study. Appendix A: The Existence and Uniqueness of Solutions. Appendix B: Theory of Determinants. Answers to Selected Problems. Index.

C. H. Edwards

Papers

The Historical Development of the Calculus

Advanced calculus of several variables

Elementary Differential Equations with Boundary Value Problems

Elementary differential equations with applications

Differential Equations & Linear Algebra