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The foundation of the modern theory of stability was created in the works of A. Poincare and A.M. Lyapunov. The theory of the stability of motion has gained increasing significance in the last decade as is apparent from the large number of publications on the subject. A considerable part of these works are concerned with practical problems, especially problems from the area of controls and servo-mechanisms, and concrete problems from engineering, which first gave the decisive impetus for the expansion and modern development of stability theory. This book contains a systematic exposition of the

Stability --- Lyapunov stability. --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Liapunov stability --- Ljapunov stability --- Control theory --- Mathematical models.

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Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmüller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrödinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

Lyapunov exponents --- Lyapunov stability --- Dynamics --- Lyapunov exponents. --- Lyapunov stability. --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Liapunov stability --- Ljapunov stability --- Control theory --- Stability --- Liapunov exponents --- Lyapunov characteristic exponents --- Differential equations

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Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. Most results are obtained in the infinite-dimensional setting of Banach spaces. Furthermore, the linear variational equations are always assumed to possess a nonuniform exponential behavior, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. The presentation is self-contained and has unified character. The volume contributes towards a rigorous mathematical foundation of the theory in the infinite-dimension setting, and may lead to further developments in the field. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.

Lyapunov stability --- Differential equations --- Calculus --- Applied Mathematics --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Lyapunov stability. --- Differential equations. --- 517.91 Differential equations --- Liapunov stability --- Ljapunov stability --- Mathematics. --- Dynamics. --- Ergodic theory. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Control theory --- Stability --- Differential Equations. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view  is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured. Various problems make the study of Lyapunov-type inequalities of interest to those in pure and applied mathematics. Originating with the study of the stability properties of the Hill equation, other questions arose for instance in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients and oscillation and intervals of disconjugacy and it lead to the study of Lyapunov-type inequalities for differential equations. This classical area of mathematics is still of great interest and remains a source of inspiration.  .

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations, Linear. --- Differential equations, Partial. --- Lyapunov stability. --- Linear differential equations --- Liapunov stability --- Ljapunov stability --- Partial differential equations --- Mathematics. --- Difference equations. --- Functional equations. --- Integral transforms. --- Operational calculus. --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Difference and Functional Equations. --- Integral Transforms, Operational Calculus. --- Linear systems --- Control theory --- Stability --- Differential Equations. --- Differential equations, partial. --- Integral Transforms. --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Equations, Functional --- Functional analysis --- 517.91 Differential equations --- Differential equations --- Operational calculus --- Electric circuits --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Calculus, Operational.

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This self-contained monograph describes basic set-theoretic methods for control and provides a discussion of their links to fundamental problems in Lyapunov stability analysis and stabilization, optimal control, control under constraints, persistent disturbance rejection, and uncertain systems analysis and synthesis. New computer technology has catalyzed a resurgence of research in this area, particularly in the development of set-theoretic techniques, many of which are computationally demanding. The work presents several established and potentially new applications, along with numerical examples and case studies. A key theme of the presentation is the trade-off between exact (but computationally intensive) and approximate (but conservative) solutions to problems. Mathematical language is kept to the minimum necessary for the adequate formulation and statement of main concepts. Numerical algorithms for the solution of the proposed problems are described in detail. Set-Theoretic Methods in Control is accessible to readers familiar with the basics of systems and control theory; prerequisites such as convexity theory are included. The text provides a solid foundation of mathematical techniques and applications and also features avenues for further theoretical study. Aimed primarily at graduate students and researchers in applied mathematics and engineering, the book will also appeal to practitioners since it contains extensive references to the literature and supplies many recipes for solving significant control problems.

Set theory --- Control theory --- Lyapunov stability --- Théorie des ensembles --- Théorie de la commande --- Liapounov, Stabilité au sens de --- Control theory. --- Engineering mathematics. --- Engineering. --- Lyapunov stability. --- Mathematical optimization. --- Set theory. --- Structural control (Engineering). --- System theory. --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Algebra --- Operations Research --- Théorie des ensembles --- Théorie de la commande --- Liapounov, Stabilité au sens de --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Liapunov stability --- Ljapunov stability --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Calculus of variations. --- Applied mathematics. --- Computational intelligence. --- Control engineering. --- Robotics. --- Mechatronics. --- Control. --- Systems Theory, Control. --- Control, Robotics, Mechatronics. --- Calculus of Variations and Optimal Control; Optimization. --- Appl.Mathematics/Computational Methods of Engineering. --- Computational Intelligence. --- Mechanical engineering --- Microelectronics --- Microelectromechanical systems --- Automation --- Machine theory --- Control engineering --- Control equipment --- Engineering instruments --- Programmable controllers --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Engineering --- Engineering analysis --- Mathematical analysis --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Systems, Theory of --- Systems science --- Science --- Construction --- Industrial arts --- Technology --- Philosophy --- Stability --- Dynamics --- Logic, Symbolic and mathematical --- Control and Systems Theory. --- Mathematical and Computational Engineering. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis --- Systems theory. --- Automation. --- Systems Theory, Control . --- Control, Robotics, Automation. --- Calculus of Variations and Optimization. --- Mathematical and Computational Engineering Applications. --- Data processing. --- Automatic factories --- Automatic production --- Computer control --- Engineering cybernetics --- Factories --- Industrial engineering --- Mechanization --- Assembly-line methods --- Automatic control --- Automatic machinery --- CAD/CAM systems --- Robotics --- Automatic control.