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The aim of this monograph is to give a thorough and self-contained account of functions of (generalized) bounded variation, the methods connected with their study, their relations to other important function classes, and their applications to various problems arising in Fourier analysis and nonlinear analysis. In the first part the basic facts about spaces of functions of bounded variation and related spaces are collected, the main ideas which are useful in studying their properties are presented, and a comparison of their importance and suitability for applications is provided, with a particular emphasis on illustrative examples and counterexamples. The second part is concerned with (sometimes quite surprising) properties of nonlinear composition and superposition operators in such spaces. Moreover, relations with Riemann-Stieltjes integrals, convergence tests for Fourier series, and applications to nonlinear integral equations are discussed. The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It is addressed to non-specialists who want to get an idea of the development of the theory and its applications in the last decades, as well as a glimpse of the diversity of the directions in which current research is moving. Since the authors try to take into account recent results and state several open problems, this book might also be a fruitful source of inspiration for further research.

Functions of bounded variation. --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Boundary Value Problem. --- Bounded Variation. --- Continuity Properties. --- Fourier Analysis. --- Monotonicity Properties. --- Nonlinear Composition Operators. --- Nonlinear Integral Equation.

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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.

Automatic control. --- Control theory. --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Dynamics --- Machine theory --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- Hermes solutions. --- Krasovskii regularization. --- Krasovskii solutions. --- Lyapunov conditions. --- Lyapunov functions. --- Lyapunov-like functions. --- asymptotic stability. --- closed sets. --- compact sets. --- conical approximation. --- conical hybrid system. --- continuity properties. --- continuous time. --- continuous-time systems. --- data structure. --- differential equations. --- differential inclusions. --- discrete time. --- discrete-time systems. --- dynamical systems. --- equilibrium points. --- flow map. --- flow set. --- generalized solutions. --- graphical convergence. --- hybrid arcs. --- hybrid control algorithms. --- hybrid dynamical systems. --- hybrid feedback control. --- hybrid models. --- hybrid system. --- hybrid time domains. --- invariance principles. --- jump map. --- jump set. --- modeling frameworks. --- modeling. --- nonlinear systems. --- numerical simulations. --- output function. --- pre-asymptotic stability. --- pre-asymptotically stable sets. --- precompact solutions. --- regularity properties. --- set convergence. --- set-valued analysis. --- set-valued mappings. --- smooth Lyapunov function. --- solution concept. --- stability theory. --- state measurement error. --- state perturbations. --- switching signals. --- switching systems. --- uniform asymptotic stability. --- well-posed hybrid systems. --- well-posed problems. --- well-posedness. --- ω-limit sets. --- Nonlinear control theory.