Control Theory in Physics and other Fields of Science
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1 Michael Schulz Control Theory in Physics and other Fields of Science Concepts, Tools, and Applications With 46 Figures Sprin ger
2 1 Introduction The Aim of Control Theory Dynamic State of Classical Mechanical Systems Dynamic State of Complex Systems What Is a Complex System? Relevant and Irrelevant Degrees of Freedom Quasi-Deterministic Versus Quasi-Stochastic Evolution The Physical Approach to Control Theory 13 References 14 2 Deterministic Control Theory Introduction: The Brachistochrone Problem The Deterministic Control Problem Functionals, Constraints, and Boundary Conditions Weak and Streng Minima The Simplest Control Problem: Classical Mechanics Euler-Lagrange Equations Optimum Criterion One-Dimensional Systems General Optimum Control Problem Lagrange Approach Hamilton Approach Pontryagin's Maximum Principle Applications of the Maximum Principle Controlled Molecular Dynamic Simulations The Hamilton-Jacobi Equation 55 References 59
3 XIV 3 Linear Quadratic Problems Introduction to Linear Quadratic Problems Motivation The Performance Functional Stability Analysis The General Solution of Linear Quadratic Problems Extensions and Applications Modifications of the Performance Inhomogeneous Linear Evolution Equations Scalar Problems The Optimal Regulator Algebraic Ricatti Equation Stability of Optimal Regulators Control of Linear Oscillations and Relaxations Integral Representation of State Dynamics Optimal Control of Generalized Linear Evolution Equations Perturbation Theory for Weakly Nonlinear Dynamics.. 88 References 90 4 Control of Fields Field Equations Classical Field Theory Hydrodynamic Field Equations Other Field Equations Control by External Sources General Aspects Control Without Spatial Boundaries Passive Boundary Conditions Control via Boundary Conditions 116 References Chaos Control Characterization of Trajectories in the Phase Space General Problems Conservative Hamiltonian Systems Nonconservative Systems Time-Discrete Chaos Control Time Continuous Control Versus Time Discrete Control Chaotic Behavior of Time Discrete Systems Control of Time Discrete Equations Reachability and Stabilizability Observability Time-Continuous Chaos Control Delayed Feedback Control 141
4 XV Synchronization 144 References Nonequilibrium Statistical Physics Statistical Approach to Phase Space Dynamics The Probability Distribution The Liouville Equation Generalized Rate Equations Probability Distribution of Relevant Quantities The Formal Solution of the Liouville Equation The Nakajima-Zwanzig Equation Notation of Probability Theory Measures of Central Tendency Measure of Fluctuations around the Central Tendency Moments and Characteristic Functions Cumulants Combined Probabilities Conditional Probability Joint Probability Markov Approximation Generalized Fokker-Planck Equation Differential Chapman-Kolmogorov Equation Deterministic Processes Markov Diffusion Processes Jump Processes Correlation and Stationarity Stationarity Correlation Spectra Stochastic Equations of Motions The Mori-Zwanzig Equation Separation of Time Scales Wiener Process Stochastic Differential Equations Ito's Formula and Fokker-Planck Equation 189 References Optimal Control of Stochastic Processes Markov Diffusion Processes under Control Information Level and Control Mechanisms Path Integrals Performance Optimal Open Loop Control Mean Performance Tree Approximation 201
5 XVI 7.3 Feedback Control The Control Equation Linear Quadratic Problems 210 References Filters and Predictors Partial Uncertainty of Controlled Systems Gaussian Processes The Central Limit Theorem Convergence Problems Levy Processes Form-Stable Limit Distributions Convergence to Stable Levy Distributions Truncated Levy Distributions Rare Events The Cramer Theorem Extreme Fluctuations Kaiman Filter Linear Quadratic Problems with Gaussian Noise Estimation of the System State Ljapunov Differential Equation Optimal Control Problem for Kaiman Filters Filters and Predictors General Filter Concepts Wiener Filters Estimation of the System Dynamics Regression and Autoregression The Bayesian Concept Neural Networks 251 References Game Theory Unpredictable Systems Optimal Control and Decision Theory Nondeterministic and Probabilistic Regime Strategies Zero-Sum Games Two-Player Games Deterministic Strategy Random Strategy Nonzero-Sum Games Nash Equilibrium Random Nash Equilibria 276 References 276
6 XVII 10 Optimization Problems Notations of Optimization Theory Introduction Convex Objects Optimization Methods Extremal Solutions Without Constraints Extremal Solutions with Constraints Linear Programming Combinatorial Optimization Problems Evolution Strategies 289 References 292
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