Title of Thesis

Open Loop Control Methods in Non-Linear Dynamical Equations

Author(s)

Babar Ahmad

Abstract
In this thesis we concentrate on open-loop control scheme to investigate the behavior of dynamical system described by non-linear differential equation. We will discuss different types of the system: classical and quantum, and apply control methods to change the dynamical behavior of the system in coordinate or momentum space.
In Chapter 1 we give some basic definitions and describe Kapitza procedure of averaging for open-loop control, then we give examples of stabilization by this method.
In Chapter 2 we propose the generalization of Kapitza procedure of averaging to stabilize the oscillator driven by periodical external field. We do this with the help of periodical external rectangular pulses, by applying Generalized Kapitza procedure of averaging, then we stabilize the oscillator by minimizing the effective potential energy function. We demonstrate the effect of stabilization with relatively low frequency for special shape of the external periodical field to compare with sin- or cos-periodical force.
In Chapter 3 an effective quantum particle beam-splitter in the momentum space is realized in the frame of open-loop control scheme. We demonstrate for small interaction time that the splitting effect §40~k with summarized relative intensity in both main components is about 50 per cent from initial intensity of the atomic beam.

1

1.1 Introduction

1.2 Mathematical Control Theory
1.3 Example of Open-Loop Control The Inverted Pendulum
1.4 Kapitza pendulum

1.5 Open-Loop Stabilization

2.1 Generalized Kapitza Method for an Arbitrary Periodical Force

2.2 Kapitza oscillator stimulated by kicking pulses
2.3 Comparison with Kapitza method

3.1 Physics Background

3.2 Mathematical Model for Beam Splitting in the Momentum Space

3.3 Shell Model for the Splitting Process Parametric Control

3.4 Numerical Simulation results for Harmonical Open-Loop control

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