| In this thesis, we address problems in control and stabilization of nonholonomic control systems arising in the areas of quantum physics, robotics, and locomotion systems. We provide a control theoretic framework for problems involving manipulation of quantum systems. The problem of design of pulse sequences in coherent spectroscopy is treated as a problem of constructive controllability in geometric control. We derive time optimal control laws for a class of control problems with drift, evolving on compact Lie groups. It is shown that these results find applications in design of pulse sequences that minimize decoherence effects in spectroscopic experiments and maximize signal to noise ratio. We also analyze in detail the problem of feedback stabilization of nonholonomic control systems. For nonholonomic systems, smooth state feedback control laws do not exist. In this thesis, we show how this topological obstruction can be overcome by embedding the system in a higher-dimensional manifold and constructing dynamic controllers. The choice of higher dimensional space is dictated by the symmetries of the system and can be interpreted as a gauge in the system. We finally draw a bridge between gauge theories in physics and control of nonholonomic control systems. |
