Optimal control and forecasting of complex dynamical systems / Ilya Grigorenko.

Format:

Book

Author/Creator:

Grigorenko, Ilya.

Contributor:

Engineering Book Fund.

Language:

English

Subjects (All):

Control theory.

Mathematical optimization.

Chaotic behavior in systems.

Differentiable dynamical systems.

Physical Description:

xiii, 198 pages : illustrations ; 24 cm

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, 2006.

Summary:

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. Book jacket.

Contents:

1 Analytical methods in control and optimization 1

1.1 Calculus of variations 1

1.1.1 The beginning: Fermat's variational principle 2

1.1.2 The "beautiful" Brachistochrone Problem 4

1.1.3 Euler-Lagrange equation 6

1.1.4 A word about distance between two functions 11

1.1.5 The Brachistochrone problem revisited 12

1.1.6 Generalizations of the Euler-Lagrange equation 14

1.1.7 Transversality conditions 16

1.1.8 Conditional extremum: Lagrange multipliers method 16

1.1.9 Mixed Optimal problem 19

1.1.10 Approximate methods of solution-Ritz's method 20

1.2 Optimal control theory 21

1.2.1 Sensitivity analysis 25

1.2.2 Null controllability 26

1.2.3 Problems with constrained control 26

2 Numerical optimization 29

2.1 The halting problem and No Free Lunch Theorem 29

2.2 Global Optimization: searching for the deepest hole on a golf field in the darkness using a cheap laser pointer 30

2.2.1 Sensitivity to numerical errors 33

2.3 Multiobjective optimization 34

2.3.1 Pareto front 35

2.3.2 The weighted-sum method 37

2.4 Simplex method 38

2.5 Simulated annealing: "crystallizing" solutions 42

2.6 Introduction to genetic algorithms 44

2.7 GA for a class of smooth (differentiable) functions 49

2.8 Application of the GA to the eigenproblem 57

2.8.1 The ground state problem in one and two dimensions 58

2.8.2 Extension of the QGA to quantum statistical problems 66

2.8.3 Formation of a "Wigner molecule" and its "melting" 69

2.9 Evolutionary gradient search and Lamarckianism 74

3 Chaos in complex systems 77

3.1 Lorenz attractor 80

3.2 Control of chaotic dynamics of the fractional Lorenz system 83

4 Optimal control of quantum systems 93

4.1 Density matrix formalism 95

4.2 Liouville equation for the reduced density matrix 96

4.3 Modern variational approach to optimal control of quantum systems 99

4.3.1 An alternative analytical theory 100

4.4 An approximate analytical solution for the case of a two level system 105

4.5 Optimal control of a time averaged occupation of the excited level in a two-level system 109

4.5.1 Analytical solution for optimal control field 114

4.5.2 Optimal control at a given time 117

4.5.3 Estimation of the absolute bound for the control due to decoherence 119

4.6 Optimal control of nanostructures: double quantum dot 121

4.6.1 The optimal field for the control of the photon assisted tunnelling between quantum dots 124

4.7 Analytical theory for control of multi-photon transitions 138

5 Optimal control and quantum computing 147

5.1 Robust two-qubit quantum registers 147

5.2 Optimal design of universal two-qubit gates 157

5.3 Entanglement of a pair of qubits 166

6 Forecasting of complex dynamical systems 171

6.1 Forecasting of financial markets 171

6.2 Autoregressive models 172

6.3 Chaos theory embedding dimensions 174

6.4 Modelling of economic "agents" and El Farol bar problem 175

6.5 Forecasting of the solar activity 176

6.6 Noise reduction and Wavelets 177

6.7 Finance and random matrix theory 179

6.8 Neural Networks 180.

Notes:

Includes bibliographical references (pages 183-196) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Engineering Book Fund.

ISBN:

9812566600

OCLC:

70142535