Some Control Problems on Lie Groups and Symmetric Spaces

Abstract

The matrix Lie groups appears naturally in physics, typically as the configuration space of some concrete mechanical problems and their corresponding dynamics leads naturally to a set of differential equations on the ambient matrix Lie group. Many authors have studied various mechanical and non-mechanical problems whose configuration space is some matrix Lie group, still there is scope for further studies. Adding to these studies, in this dissertation, we have defined control systems on some mechanical problems, on Lie groups which play important role in physics and control systems defined on the groups associated with the Black-Scholes equation, which is an integral part of finance. This dissertation presents an introduction to nonlinear control systems, with main emphasis on controllability properties of such systems. We have analyzed the controllability of the systems, using certain Lie algebras of the vector fields defined by the system.
This dissertation studies the system dynamics and optimal control problems by providing theoretical analyses while considering the fundamental geometric properties and by using computational algorithms that preserve those geometric properties. The optimal control problems of systems like, kinematic car and kinematic unicycle model, with a Lie group configuration manifold is discussed. Optimal control problems for systems defined on matrix Lie groups SU(2), S0(3; 1) and a group associated with the conformal coordinate transformations of Black-Scholes equation is also considered. Further more, two unconventional integrators are applied to the system dynamics and some geometric properties are discussed. The trajectories for both the unconventional integrators and conventional Runge-Kutta integrator, for each of the control systems is shown.
In this dissertation, the study of mathematical control theory on a manifold which is simultaneously a symmetric space is discussed from a geometric viewpoint. There has been many studies and theories developed in a manifold, i.e., Lie groups. But in recent times, many new kinetic models have been developed with the help of exponential submanifolds in the areas of robotics, control, etc. It thus becomes necessary to study the behavior of the control system defined on the exponential submanifolds, also known as symmetric spaces. There are many parallel similarities between the theory of Lie groups and symmetric spaces. Lie triple system and symmetric spaces are related by exponential mapping similar to Lie groups and Lie algebras. As a result, parallel theories and calculations of control theory on Lie groups and Lie algebras can be extended to the case of symmetric spaces and Lie triple systems. The first natural problem to study in control theory is the controllability problem,i.e., to characterize the states reachable from a given initial state. So global controllability condition for system defined on symmetric spaces is established and illustrated by few examples of controllable systems of exponential submanifolds of SE ( 3) and random matrix ensembles