Geometric Control



Linear algebra, Advanced calculus, Differential geometry, (Naive) Lie theory. Some exposure to control theory and/or mechanics would be advantageous.


The main objective is to familiarize the students with the powerful (differential) geometric methods used in modern optimal control theory. Control theory deals with (dynamical) systems that can be controlled. It became recognized as a mathematical subject in the 1960's and since the mid 1970's one can speak of a "(differential) geometric trend" in nonlinear control. Geometrically, a control system is a family of vector fields parametrized by controls. Modern optimal control theory can be seen as a generalization of the classical calculus of variations to systems with nonholonomoc constraints. An extremely powerful tool in dealing with optimal control problems is the celebrated Pontryagin's maximum principle. Our approach is "concrete" in the sense that we confine ourselves to optimal control problems on matrix Lie groups.


Lie groups and Lie algebras, left-invariant control systems, extension techniques (for left-invariant systems), induced systems on homogeneous spaces, controllability, Pontryagin's maximum principle, examples of optimal control problems on Lie groups.


''Control Theory on Lie Groups" by Sachkov ( SISSA Lecture Notes, 2006).


ONE tutorial per week.


TWO tests and ONE final examination.

Last Modified: Fri, 15 Jul 2011 14:57:09 SAST