| Introduction | p. 1 |
| The Aim of Control Theory | p. 1 |
| Dynamic State of Classical Mechanical Systems | p. 3 |
| Dynamic State of Complex Systems | p. 6 |
| What Is a Complex System? | p. 6 |
| Relevant and Irrelevant Degrees of Freedom | p. 9 |
| Quasi-Deterministic Versus Quasi-Stochastic Evolution | p. 10 |
| The Physical Approach to Control Theory | p. 13 |
| References | p. 14 |
| Deterministic Control Theory | p. 17 |
| Introduction: The Brachistochrone Problem | p. 17 |
| The Deterministic Control Problem | p. 19 |
| Functionals, Constraints, and Boundary Conditions | p. 19 |
| Weak and Strong Minima | p. 20 |
| The Simplest Control Problem: Classical Mechanics | p. 22 |
| Euler-Lagrange Equations | p. 22 |
| Optimum Criterion | p. 24 |
| One-Dimensional Systems | p. 30 |
| General Optimum Control Problem | p. 33 |
| Lagrange Approach | p. 33 |
| Hamilton Approach | p. 40 |
| Pontryagin's Maximum Principle | p. 42 |
| Applications of the Maximum Principle | p. 45 |
| Controlled Molecular Dynamic Simulations | p. 53 |
| The Hamilton-Jacobi Equation | p. 55 |
| References | p. 59 |
| Linear Quadratic Problems | p. 61 |
| Introduction to Linear Quadratic Problems | p. 61 |
| Motivation | p. 61 |
| The Performance Functional | p. 62 |
| Stability Analysis | p. 63 |
| The General Solution of Linear Quadratic Problems | p. 71 |
| Extensions and Applications | p. 73 |
| Modifications of the Performance | p. 73 |
| Inhomogeneous Linear Evolution Equations | p. 75 |
| Scalar Problems | p. 75 |
| The Optimal Regulator | p. 77 |
| Algebraic Ricatti Equation | p. 77 |
| Stability of Optimal Regulators | p. 79 |
| Control of Linear Oscillations and Relaxations | p. 81 |
| Integral Representation of State Dynamics | p. 81 |
| Optimal Control of Generalized Linear Evolution Equations | p. 85 |
| Perturbation Theory for Weakly Nonlinear Dynamics | p. 88 |
| References | p. 90 |
| Control of Fields | p. 93 |
| Field Equations | p. 93 |
| Classical Field Theory | p. 93 |
| Hydrodynamic Field Equations | p. 99 |
| Other Field Equations | p. 101 |
| Control by External Sources | p. 103 |
| General Aspects | p. 103 |
| Control Without Spatial Boundaries | p. 104 |
| Passive Boundary Conditions | p. 114 |
| Control via Boundary Conditions | p. 116 |
| References | p. 118 |
| Chaos Control | p. 123 |
| Characterization of Trajectories in the Phase Space | p. 123 |
| General Problems | p. 123 |
| Conservative Hamiltonian Systems | p. 124 |
| Nonconservative Systems | p. 126 |
| Time-Discrete Chaos Control | p. 128 |
| Time Continuous Control Versus Time Discrete Control | p. 128 |
| Chaotic Behavior of Time Discrete Systems | p. 132 |
| Control of Time Discrete Equations | p. 135 |
| Reachability and Stabilizability | p. 137 |
| Observability | p. 140 |
| Time-Continuous Chaos Control | p. 141 |
| Delayed Feedback Control | p. 141 |
| Synchronization | p. 144 |
| References | p. 146 |
| Nonequilibrium Statistical Physics | p. 149 |
| Statistical Approach to Phase Space Dynamics | p. 149 |
| The Probability Distribution | p. 149 |
| The Liouville Equation | p. 152 |
| Generalized Rate Equations | p. 153 |
| Probability Distribution of Relevant Quantities | p. 153 |
| The Formal Solution of the Liouville Equation | p. 155 |
| The Nakajima-Zwanzig Equation | p. 156 |
| Notation of Probability Theory | p. 161 |
| Measures of Central Tendency | p. 161 |
| Measure of Fluctuations around the Central Tendency | p. 162 |
| Moments and Characteristic Functions | p. 162 |
| Cumulants | p. 163 |
| Combined Probabilities | p. 164 |
| Conditional Probability | p. 164 |
| Joint Probability | p. 165 |
| Markov Approximation | p. 167 |
| Generalized Fokker-Planck Equation | p. 169 |
| Differential Chapman-Kolmogorov Equation | p. 169 |
| Deterministic Processes | p. 173 |
| Markov Diffusion Processes | p. 174 |
| Jump Processes | p. 175 |
| Correlation and Stationarity | p. 176 |
| Stationarity | p. 176 |
| Correlation | p. 177 |
| Spectra | p. 178 |
| Stochastic Equations of Motions | p. 179 |
| The Mori-Zwanzig Equation | p. 179 |
| Separation of Time Scales | p. 182 |
| Wiener Process | p. 183 |
| Stochastic Differential Equations | p. 185 |
| Ito's Formula and Fokker-Planck Equation | p. 189 |
| References | p. 191 |
| Optimal Control of Stochastic Processes | p. 193 |
| Markov Diffusion Processes under Control | p. 193 |
| Information Level and Control Mechanisms | p. 193 |
| Path Integrals | p. 194 |
| Performance | p. 197 |
| Optimal Open Loop Control | p. 199 |
| Mean Performance | p. 199 |
| Tree Approximation | p. 201 |
| Feedback Control | p. 204 |
| The Control Equation | p. 204 |
| Linear Quadratic Problems | p. 210 |
| References | p. 211 |
| Filters and Predictors | p. 213 |
| Partial Uncertainty of Controlled Systems | p. 213 |
| Gaussian Processes | p. 215 |
| The Central Limit Theorem | p. 215 |
| Convergence Problems | p. 220 |
| Lévy Processes | p. 223 |
| Form-Stable Limit Distributions | p. 223 |
| Convergence to Stable Lévy Distributions | p. 226 |
| Truncated Lévy Distributions | p. 227 |
| Rare Events | p. 228 |
| The Cramér Theorem | p. 228 |
| Extreme Fluctuations | p. 230 |
| Kalman Filter | p. 232 |
| Linear Quadratic Problems with Gaussian Noise | p. 232 |
| Estimation of the System State | p. 232 |
| Ljapunov Differential Equation | p. 237 |
| Optimal Control Problem for Kalman Filters | p. 239 |
| Filters and Predictors | p. 243 |
| General Filter Concepts | p. 243 |
| Wiener Filters | p. 244 |
| Estimation of the System Dynamics | p. 245 |
| Regression and Autoregression | p. 246 |
| The Bayesian Concept | p. 249 |
| Neural Networks | p. 251 |
| References | p. 261 |
| Game Theory | p. 265 |
| Unpredictable Systems | p. 265 |
| Optimal Control and Decision Theory | p. 267 |
| Nondeterministic and Probabilistic Regime | p. 267 |
| Strategies | p. 269 |
| Zero-Sum Games | p. 271 |
| Two-Player Games | p. 271 |
| Deterministic Strategy | p. 272 |
| Random Strategy | p. 273 |
| Nonzero-Sum Games | p. 274 |
| Nash Equilibrium | p. 274 |
| Random Nash Equilibria | p. 276 |
| References | p. 276 |
| Optimization Problems | p. 279 |
| Notations of Optimization Theory | p. 279 |
| Introduction | p. 279 |
| Convex Objects | p. 280 |
| Optimization Methods | p. 282 |
| Extremal Solutions Without Constraints | p. 282 |
| Extremal Solutions with Constraints | p. 285 |
| Linear Programming | p. 286 |
| Combinatorial Optimization Problems | p. 287 |
| Evolution Strategies | p. 289 |
| References | p. 292 |
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