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Optimal Control And Forecasting Of Complex Dynamical Systems

By: Ilya Grigorenko

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Hardcover | 1 May 2006

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212 Pages

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23.32 x 15.65 x 1.96

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This important book reviews applications of optimization and optimal control theory to modern problems in physics, nano-science and finance. The theory presented here can be efficiently applied to various problems, such as the determination of the optimal shape of a laser pulse to induce certain excitations in quantum systems, the optimal design of nanostructured materials and devices, or the control of chaotic systems and minimization of the forecast error for a given forecasting model (for example, artificial neural networks). Starting from a brief review of the history of variational calculus, the book discusses optimal control theory and global optimization using modern numerical techniques. Key elements of chaos theory and basics of fractional derivatives, which are useful in control and forecast of complex dynamical systems, are presented. The coverage includes several interdisciplinary problems to demonstrate the efficiency of the presented algorithms, and different methods of forecasting complex dynamics are discussed. Contents: Analytical Methods in Control and Optimization; Numerical Optimization; Chaos in Complex Systems; Optimal Control of Quantum Systems; Optimal Control and Quantum Computing; Forecasting of Complex Dynamical Systems. Key Feature Gives a broad interdisciplinary approach (optimal control, genetic algorithms, chaos theory, wavelets, fractional derivatives) to solve different optimization and forecasting problems with practical examples Readership: Graduate students and researchers in optimal control, numerical optimization and forecasting of complex systems.

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Chaos Theory Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids

Dynamics & Vibration Optimisation Calculus & Mathematical Analysis Differential Calculus & Equations

Preface	p. vii
Analytical methods in control and optimization	p. 1
Calculus of variations	p. 1
The beginning: Fermat's variational principle	p. 2
The "beautiful" Brachistochrone Problem	p. 4
Euler-Lagrange equation	p. 6
A word about distance between two functions	p. 11
The Brachistochrone problem revisited	p. 12
Generalizations of the Euler-Lagrange equation	p. 14
Transversality conditions	p. 16
Conditional extremum: Lagrange multipliers method	p. 16
Mixed Optimal problem	p. 19
Approximate methods of solution-Ritz's method	p. 20
Optimal control theory	p. 21
Sensitivity analysis	p. 25
Null controllability	p. 26
Problems with constrained control	p. 26
Summary	p. 27
Numerical optimization	p. 29
The halting problem and No Free Lunch Theorem	p. 29
Global Optimization: searching for the deepest hole on a golf field in the darkness using a cheap laser pointer	p. 30
Sensitivity to numerical errors	p. 33
Multiobjective optimization	p. 34
Pareto front	p. 35
The weighted-sum method	p. 37
Simplex method	p. 38
Simulated annealing: "crystallizing" solutions	p. 42
Introduction to genetic algorithms	p. 44
GA for a class of smooth (differentiable) functions	p. 49
Application of the GA to the eigenproblem	p. 57
The ground state problem in one and two dimensions	p. 58
Extension of the QGA to quantum statistical problems	p. 66
Formation of a "Wigner molecule" and its "melting"	p. 69
Evolutionary gradient search and Lamarckianism	p. 74
Summary	p. 76
Chaos in complex systems	p. 77
Lorenz attractor	p. 80
Control of chaotic dynamics of the fractional Lorenz system	p. 83
Summary	p. 91
Optimal control of quantum systems	p. 93
Density matrix formalism	p. 95
Liouville equation for the reduced density matrix	p. 96
Modern variational approach to optimal control of quantum systems	p. 99
An alternative analytical theory	p. 100
An approximate analytical solution for the case of a two level system	p. 105
Optimal control of a time averaged occupation of the excited level in a two-level system	p. 109
Analytical solution for optimal control field	p. 114
Optimal control at a given time	p. 117
Estimation of the absolute bound for the control due to decoherence	p. 119
Optimal control of nanostructures: double quantum dot	p. 121
The optimal field for the control of the photon assisted tunnelling between quantum dots	p. 124
Analytical theory for control of multi-photon transitions	p. 138
Summary	p. 145
Optimal control and quantum computing	p. 147
Robust two-qubit quantum registers	p. 147
Optimal design of universal two-qubit gates	p. 157
Entanglement of a pair of qubits	p. 166
Summary	p. 168
Forecasting of complex dynamical systems	p. 171
Forecasting of financial markets	p. 171
Autoregressive models	p. 172
Chaos theory embedding dimensions	p. 174
Modelling of economic "agents" and El Farol bar problem	p. 175
Forecasting of the solar activity	p. 176
Noise reduction and Wavelets	p. 177
Finance and random matrix theory	p. 179
Neural Networks	p. 180
Summary	p. 181
Bibliography	p. 183
Index	p. 197
Table of Contents provided by Ingram. All Rights Reserved.

Optimal Control And Forecasting Of Complex Dynamical Systems

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Hardcover

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