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

Format:

Book

Author/Creator:

Grigorenko, Ilya.

Language:

English

Subjects (All):

Chaotic behavior in systems.

Control theory.

Differentiable dynamical systems.

Mathematical optimization.

Physical Description:

1 online resource (213 p.)

Place of Publication:

Hackensack, NJ : World Scientific, c2006.

Language Note:

English

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 calcul

Contents:

Preface; Contents; Chapter 1. Analytical methods in control and optimization; 1.1 Calculus of variations; 1.1.1 The beginning: Fermat's variational principle; 1.1.2 The ""beautiful"" Brachistochrone Problem; 1.1.3 Euler-Lagrange equation; 1.1.4 A word about distance between two functions; 1.1.5 The Brachistochrone problem revisited; 1.1.6 Generalizations of the Euler-Lagrange equation; 1.1.7 Transversality conditions; 1.1.8 Conditional extremum: Lagrange multipliers method; 1.1.9 Mixed Optimal problem; 1.1.10 Approximate methods of solution-Ritz's method; 1.2 Optimal control theory

1.2.1 Sensitivity analysis1.2.2 Null controllability; 1.2.3 Problems with constrained control; 1.3 Summary; Chapter 2. Numerical optimization; 2.1 The halting problem and No Free Lunch Theorem; 2.2 Global Optimization: searching for the deepest hole on a golf field in the darkness using a cheap laser pointer; 2.2.1 Sensitivity to numerical errors; 2.3 Multiobjective optimization; 2.3.1 Pareto front; 2.3.2 The weighted-sum method; 2.4 Simplex method; 2.5 Simulated annealing: ""crystallizing"" solutions; 2.6 Introduction to genetic algorithms

2.7 GA for a class of smooth (differentiable) functions2.8 Application of the GA to the eigenproblem; 2.8.1 The ground state problem in one and two dimensions; 2.8.2 Extension of the QGA to quantum statistical problems; 2.8.3 Formation of a ""Wigner molecule"" and its ""melting""; 2.9 Evolutionary gradient search and Lamarckianism; 2.10 Summary; Chapter 3. Chaos in complex systems; 3.1 Lorenz attractor; 3.2 Control of chaotic dynamics of the fractional Lorenz system; 3.3 Summary; Chapter 4. Optimal control of quantum systems; 4.1 Density matrix formalism

4.2 Liouville equation for the reduced density matrix4.3 Modern variational approach to optimal control of quantum systems; 4.3.1 An alternative analytical theory; 4.4 An approximate analytical solution for the case of a two level system; 4.5 Optimal control of a time averaged occupation of the excited level in a two-level system; 4.5.1 Analytical solution for optimal control field; 4.5.2 Optimal control at a given time; 4.5.3 Estimation of the absolute bound for the control due to decoherence; 4.6 Optimal control of nanostructures: double quantum dot

4.6.1 The optimal field for the control of the photon assisted tunnelling between quantum dots4.7 Analytical theory for control of multi-photon transitions; 4.8 Summary; Chapter 5. Optimal control and quantum computing; 5.1 Robust two-qubit quantum registers; 5.2 Optimal design of universal two-qubit gates; 5.3 Entanglement of a pair of qubits; 5.4 Summary; Chapter 6. Forecasting of complex dynamical systems; 6.1 Forecasting of financial markets; 6.2 Autoregressive models; 6.3 Chaos theory embedding dimensions; 6.4 Modelling of economic ""agents"" and El Farol bar problem

6.5 Forecasting of the solar activity

Notes:

Description based upon print version of record.

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

ISBN:

9786611919368

9781281919366

1281919365

9789812774248

9812774246