Optimal Control And Forecasting of Complex Dynamical Systems

by Grigorenko, Ilya

• ISBN13: 9789812566607
• ISBN10: 9812566600
• Format: Hardcover
• Copyright: 2006-04-30
• Publisher: World Scientific Pub Co Inc

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. 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.

Table of Contents

Preface	vii
1. Analytical methods in control and optimization	1	(28)
1.1 Calculus of variations	1	(20)
1.1.1 The beginning: Fermat's variational principle	2	(2)
1.1.2 The "beautiful" Brachistochrone Problem	4	(2)
1.1.3 Euler-Lagrange equation	6	(5)
1.1.4 A word about distance between two functions	11	(1)
1.1.5 The Brachistochrone problem revisited	12	(2)
1.1.6 Generalizations of the Euler-Lagrange equation	14	(2)
1.1.7 Transversality conditions	16	(1)
1.1.8 Conditional extremum: Lagrange multipliers method	16	(3)
1.1.9 Mixed Optimal problem	19	(1)
1.1.10 Approximate methods of solution-Ritz's method	20	(1)
1.2 Optimal control theory	21	(6)
1.2.1 Sensitivity analysis	25	(1)
1.2.2 Null controllability	26	(1)
1.2.3 Problems with constrained control	26	(1)
1.3 Summary	27	(2)
2. Numerical optimization	29	(48)
2.1 The halting problem and No Free Lunch Theorem	29	(1)
2.2 Global Optimization: searching for the deepest hole on a golf field in the darkness using a cheap laser pointer	30	(4)
2.2.1 Sensitivity to numerical errors	33	(1)
2.3 Multiobjective optimization	34	(4)
2.3.1 Pareto front	35	(2)
2.3.2 The weighted-sum method	37	(1)
2.4 Simplex method	38	(4)
2.5 Simulated annealing: "crystallizing" solutions	42	(2)
2.6 Introduction to genetic algorithms	44	(5)
2.7 GA for a class of smooth (differentiable) functions	49	(8)
2.8 Application of the GA to the eigenproblem	57	(17)
2.8.1 The ground state problem in one and two dimensions	58	(8)
2.8.2 Extension of the QGA to quantum statistical problems	66	(3)
2.8.3 Formation of a "Wigner molecule" and its "melting"	69	(5)
2.9 Evolutionary gradient search and Lamarckianism	74	(2)
2.10 Summary	76	(1)
3. Chaos in complex systems	77	(16)
3.1 Lorenz attractor	80	(3)
3.2 Control of chaotic dynamics of the fractional Lorenz system	83	(8)
3.3 Summary	91	(2)
4. Optimal control of quantum systems	93	(54)
4.1 Density matrix formalism	95	(1)
4.2 Liouville equation for the reduced density matrix	96	(3)
4.3 Modern variational approach to optimal control of quantum systems	99	(6)
4.3.1 An alternative analytical theory	100	(5)
4.4 An approximate analytical solution for the case of a two level system	105	(4)
4.5 Optimal control of a time averaged occupation of the excited level in a two-level system	109	(12)
4.5.1 Analytical solution for optimal control field	114	(3)
4.5.2 Optimal control at a given time	117	(2)
4.5.3 Estimation of the absolute bound for the control due to decoherence	119	(2)
4.6 Optimal control of nanostructures: double quantum dot	121	(17)
4.6.1 The optimal field for the control of the photon assisted tunnelling between quantum dots	124	(14)
4.7 Analytical theory for control of multi-photon transitions	138	(7)
4.8 Summary	145	(2)
5. Optimal control and quantum computing	147	(24)
5.1 Robust two-qubit quantum registers	147	(10)
5.2 Optimal design of universal two-qubit gates	157	(9)
5.3 Entanglement of a pair of qubits	166	(2)
5.4 Summary	168	(3)
6. Forecasting of complex dynamical systems	171	(12)
6.1 Forecasting of financial markets	171	(1)
6.2 Autoregressive models	172	(2)
6.3 Chaos theory embedding dimensions	174	(1)
6.4 Modelling of economic "agents" and El Farol bar problem	175	(1)
6.5 Forecasting of the solar activity	176	(1)
6.6 Noise reduction and Wavelets	177	(2)
6.7 Finance and random matrix theory	179	(1)
6.8 Neural Networks	180	(1)
6.9 Summary	181	(2)
Bibliography	183	(14)
Index	197