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This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Schrödinger operator. --- Green's functions. --- Hamiltonian systems. --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Operator, Schrödinger --- Differential equations --- Differentiable dynamical systems --- Potential theory (Mathematics) --- Differential operators --- Quantum theory --- Schrödinger equation --- Almost Mathieu operator. --- Analytic function. --- Anderson localization. --- Betti number. --- Cartan's theorem. --- Chaos theory. --- Density of states. --- Dimension (vector space). --- Diophantine equation. --- Dynamical system. --- Equation. --- Existential quantification. --- Fundamental matrix (linear differential equation). --- Green's function. --- Hamiltonian system. --- Hermitian adjoint. --- Infimum and supremum. --- Iterative method. --- Jacobi operator. --- Linear equation. --- Linear map. --- Linearization. --- Monodromy matrix. --- Non-perturbative. --- Nonlinear system. --- Normal mode. --- Parameter space. --- Parameter. --- Parametrization. --- Partial differential equation. --- Periodic boundary conditions. --- Phase space. --- Phase transition. --- Polynomial. --- Renormalization. --- Self-adjoint. --- Semialgebraic set. --- Special case. --- Statistical significance. --- Subharmonic function. --- Summation. --- Theorem. --- Theory. --- Transfer matrix. --- Transversality (mathematics). --- Trigonometric functions. --- Trigonometric polynomial. --- Uniformization theorem.

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This book develops a general analysis and synthesis framework for impulsive and hybrid dynamical systems. Such a framework is imperative for modern complex engineering systems that involve interacting continuous-time and discrete-time dynamics with multiple modes of operation that place stringent demands on controller design and require implementation of increasing complexity--whether advanced high-performance tactical fighter aircraft and space vehicles, variable-cycle gas turbine engines, or air and ground transportation systems. Impulsive and Hybrid Dynamical Systems goes beyond similar treatments by developing invariant set stability theorems, partial stability, Lagrange stability, boundedness, ultimate boundedness, dissipativity theory, vector dissipativity theory, energy-based hybrid control, optimal control, disturbance rejection control, and robust control for nonlinear impulsive and hybrid dynamical systems. A major contribution to mathematical system theory and control system theory, this book is written from a system-theoretic point of view with the highest standards of exposition and rigor. It is intended for graduate students, researchers, and practitioners of engineering and applied mathematics as well as computer scientists, physicists, and other scientists who seek a fundamental understanding of the rich dynamical behavior of impulsive and hybrid dynamical systems.

Automatic control. --- Control theory. --- Dynamics. --- Discrete-time systems. --- 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 --- DES (System analysis) --- Discrete event systems --- Sampled-data systems --- Digital control systems --- Discrete mathematics --- System analysis --- Linear time invariant systems --- Actuator. --- Adaptive control. --- Algorithm. --- Amplitude. --- Analog computer. --- Arbitrarily large. --- Asymptote. --- Asymptotic analysis. --- Axiom. --- Balance equation. --- Bode plot. --- Boundedness. --- Calculation. --- Center of mass (relativistic). --- Coefficient of restitution. --- Continuous function. --- Convex set. --- Differentiable function. --- Differential equation. --- Dissipation. --- Dissipative system. --- Dynamical system. --- Dynamical systems theory. --- Energy. --- Equations of motion. --- Equilibrium point. --- Escapement. --- Euler–Lagrange equation. --- Exponential stability. --- Forms of energy. --- Hamiltonian mechanics. --- Hamiltonian system. --- Hermitian matrix. --- Hooke's law. --- Hybrid system. --- Identity matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Initial condition. --- Instability. --- Interconnection. --- Invariance theorem. --- Isolated system. --- Iterative method. --- Jacobian matrix and determinant. --- Lagrangian (field theory). --- Lagrangian system. --- Lagrangian. --- Likelihood-ratio test. --- Limit cycle. --- Limit set. --- Linear function. --- Linearization. --- Lipschitz continuity. --- Lyapunov function. --- Lyapunov stability. --- Mass balance. --- Mathematical optimization. --- Melting. --- Mixture. --- Moment of inertia. --- Momentum. --- Monotonic function. --- Negative feedback. --- Nonlinear programming. --- Nonlinear system. --- Nonnegative matrix. --- Optimal control. --- Ordinary differential equation. --- Orthant. --- Parameter. --- Partial differential equation. --- Passive dynamics. --- Poincaré conjecture. --- Potential energy. --- Proof mass. --- Quantity. --- Rate function. --- Requirement. --- Robust control. --- Second law of thermodynamics. --- Semi-infinite. --- Small-gain theorem. --- Special case. --- Spectral radius. --- Stability theory. --- State space. --- Stiffness. --- Supply (economics). --- Telecommunication. --- Theorem. --- Transpose. --- Uncertainty. --- Uniform boundedness. --- Uniqueness. --- Vector field. --- Vibration. --- Zeroth (software). --- Zeroth law of thermodynamics.