Control theory in physics and other fields of science : concepts, tools, and applications

Title:

Personal Author:

Schulz, Michael, 1959-

Series:

Springer tracts in modern physics ; 215

Publication Information:

Berlin : Springer, 2006

ISBN:

9783540295143

Subject Term:

Control theory

Science -- Mathematical models

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010106900	QC1 S33 2006	Open Access Book	Book	Searching... Unknown
Searching... PSZ KL	30000010113175	QC1 S33 2006	Open Access Book	Book	Searching... Unknown

This book provides an introduction to the analysis and the control mechanism of physical, chemical, biological, technological and economic models and their nonequilibrium evolution dynamics. Strong emphasis is placed on the foundation of variational principles, evolution and control equations, numerical methods, statistical concepts and techniques for solving or estimation of stochastic control problems for systems with a high degree of complexity. In particular, the central aim of this book is developing a synergetic connection between theoretical concepts and real applications. This book is a modern introduction and a helpful tool for researchers as well as for graduate students interested in econophysics and related topics.

1 Introduction	p. 1
1.1 The Aim of Control Theory	p. 1
1.2 Dynamic State of Classical Mechanical Systems	p. 3
1.3 Dynamic State of Complex Systems	p. 6
1.3.1 What Is a Complex System?	p. 6
1.3.2 Relevant and Irrelevant Degrees of Freedom	p. 9
1.3.3 Quasi-Deterministic Versus Quasi-Stochastic Evolution	p. 10
1.4 The Physical Approach to Control Theory	p. 13
References	p. 14
2 Deterministic Control Theory	p. 17
2.1 Introduction: The Brachistochrone Problem	p. 17
2.2 The Deterministic Control Problem	p. 19
2.2.1 Functionals, Constraints, and Boundary Conditions	p. 19
2.2.2 Weak and Strong Minima	p. 20
2.3 The Simplest Control Problem: Classical Mechanics	p. 22
2.3.1 Euler-Lagrange Equations	p. 22
2.3.2 Optimum Criterion	p. 24
2.3.3 One-Dimensional Systems	p. 30
2.4 General Optimum Control Problem	p. 33
2.4.1 Lagrange Approach	p. 33
2.4.2 Hamilton Approach	p. 40
2.4.3 Pontryagin's Maximum Principle	p. 42
2.4.4 Applications of the Maximum Principle	p. 45
2.4.5 Controlled Molecular Dynamic Simulations	p. 53
2.5 The Hamilton-Jacobi Equation	p. 55
References	p. 59
3 Linear Quadratic Problems	p. 61
3.1 Introduction to Linear Quadratic Problems	p. 61
3.1.1 Motivation	p. 61
3.1.2 The Performance Functional	p. 62
3.1.3 Stability Analysis	p. 63
3.1.4 The General Solution of Linear Quadratic Problems	p. 71
3.2 Extensions and Applications	p. 73
3.2.1 Modifications of the Performance	p. 73
3.2.2 Inhomogeneous Linear Evolution Equations	p. 75
3.2.3 Scalar Problems	p. 75
3.3 The Optimal Regulator	p. 77
3.3.1 Algebraic Ricatti Equation	p. 77
3.3.2 Stability of Optimal Regulators	p. 79
3.4 Control of Linear Oscillations and Relaxations	p. 81
3.4.1 Integral Representation of State Dynamics	p. 81
3.4.2 Optimal Control of Generalized Linear Evolution Equations	p. 85
3.4.3 Perturbation Theory for Weakly Nonlinear Dynamics	p. 88
References	p. 90
4 Control of Fields	p. 93
4.1 Field Equations	p. 93
4.1.1 Classical Field Theory	p. 93
4.1.2 Hydrodynamic Field Equations	p. 99
4.1.3 Other Field Equations	p. 101
4.2 Control by External Sources	p. 103
4.2.1 General Aspects	p. 103
4.2.2 Control Without Spatial Boundaries	p. 104
4.2.3 Passive Boundary Conditions	p. 114
4.3 Control via Boundary Conditions	p. 116
References	p. 118
5 Chaos Control	p. 123
5.1 Characterization of Trajectories in the Phase Space	p. 123
5.1.1 General Problems	p. 123
5.1.2 Conservative Hamiltonian Systems	p. 124
5.1.3 Nonconservative Systems	p. 126
5.2 Time-Discrete Chaos Control	p. 128
5.2.1 Time Continuous Control Versus Time Discrete Control	p. 128
5.2.2 Chaotic Behavior of Time Discrete Systems	p. 132
5.2.3 Control of Time Discrete Equations	p. 135
5.2.4 Reachability and Stabilizability	p. 137
5.2.5 Observability	p. 140
5.3 Time-Continuous Chaos Control	p. 141
5.3.1 Delayed Feedback Control	p. 141
5.3.2 Synchronization	p. 144
References	p. 146
6 Nonequilibrium Statistical Physics	p. 149
6.1 Statistical Approach to Phase Space Dynamics	p. 149
6.1.1 The Probability Distribution	p. 149
6.2 The Liouville Equation	p. 152
6.3 Generalized Rate Equations	p. 153
6.3.1 Probability Distribution of Relevant Quantities	p. 153
6.3.2 The Formal Solution of the Liouville Equation	p. 155
6.3.3 The Nakajima-Zwanzig Equation	p. 156
6.4 Notation of Probability Theory	p. 161
6.4.1 Measures of Central Tendency	p. 161
6.4.2 Measure of Fluctuations around the Central Tendency	p. 162
6.4.3 Moments and Characteristic Functions	p. 162
6.4.4 Cumulants	p. 163
6.5 Combined Probabilities	p. 164
6.5.1 Conditional Probability	p. 164
6.5.2 Joint Probability	p. 165
6.6 Markov Approximation	p. 167
6.7 Generalized Fokker-Planck Equation	p. 169
6.7.1 Differential Chapman-Kolmogorov Equation	p. 169
6.7.2 Deterministic Processes	p. 173
6.7.3 Markov Diffusion Processes	p. 174
6.7.4 Jump Processes	p. 175
6.8 Correlation and Stationarity	p. 176
6.8.1 Stationarity	p. 176
6.8.2 Correlation	p. 177
6.8.3 Spectra	p. 178
6.9 Stochastic Equations of Motions	p. 179
6.9.1 The Mori-Zwanzig Equation	p. 179
6.9.2 Separation of Time Scales	p. 182
6.9.3 Wiener Process	p. 183
6.9.4 Stochastic Differential Equations	p. 185
6.9.5 Ito's Formula and Fokker-Planck Equation	p. 189
References	p. 191
7 Optimal Control of Stochastic Processes	p. 193
7.1 Markov Diffusion Processes under Control	p. 193
7.1.1 Information Level and Control Mechanisms	p. 193
7.1.2 Path Integrals	p. 194
7.1.3 Performance	p. 197
7.2 Optimal Open Loop Control	p. 199
7.2.1 Mean Performance	p. 199
7.2.2 Tree Approximation	p. 201
7.3 Feedback Control	p. 204
7.3.1 The Control Equation	p. 204
7.3.2 Linear Quadratic Problems	p. 210
References	p. 211
8 Filters and Predictors	p. 213
8.1 Partial Uncertainty of Controlled Systems	p. 213
8.2 Gaussian Processes	p. 215
8.2.1 The Central Limit Theorem	p. 215
8.2.2 Convergence Problems	p. 220
8.3 Lévy Processes	p. 223
8.3.1 Form-Stable Limit Distributions	p. 223
8.3.2 Convergence to Stable Lévy Distributions	p. 226
8.3.3 Truncated Lévy Distributions	p. 227
8.4 Rare Events	p. 228
8.4.1 The Cramér Theorem	p. 228
8.4.2 Extreme Fluctuations	p. 230
8.5 Kalman Filter	p. 232
8.5.1 Linear Quadratic Problems with Gaussian Noise	p. 232
8.5.2 Estimation of the System State	p. 232
8.5.3 Ljapunov Differential Equation	p. 237
8.5.4 Optimal Control Problem for Kalman Filters	p. 239
8.6 Filters and Predictors	p. 243
8.6.1 General Filter Concepts	p. 243
8.6.2 Wiener Filters	p. 244
8.6.3 Estimation of the System Dynamics	p. 245
8.6.4 Regression and Autoregression	p. 246
8.6.5 The Bayesian Concept	p. 249
8.6.6 Neural Networks	p. 251
References	p. 261
9 Game Theory	p. 265
9.1 Unpredictable Systems	p. 265
9.2 Optimal Control and Decision Theory	p. 267
9.2.1 Nondeterministic and Probabilistic Regime	p. 267
9.2.2 Strategies	p. 269
9.3 Zero-Sum Games	p. 271
9.3.1 Two-Player Games	p. 271
9.3.2 Deterministic Strategy	p. 272
9.3.3 Random Strategy	p. 273
9.4 Nonzero-Sum Games	p. 274
9.4.1 Nash Equilibrium	p. 274
9.4.2 Random Nash Equilibria	p. 276
References	p. 276
10 Optimization Problems	p. 279
10.1 Notations of Optimization Theory	p. 279
10.1.1 Introduction	p. 279
10.1.2 Convex Objects	p. 280
10.2 Optimization Methods	p. 282
10.2.1 Extremal Solutions Without Constraints	p. 282
10.2.2 Extremal Solutions with Constraints	p. 285
10.2.3 Linear Programming	p. 286
10.2.4 Combinatorial Optimization Problems	p. 287
10.2.5 Evolution Strategies	p. 289
References	p. 292

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