Course Development

This is a two-semester introduction to dynamical systems, nonlinear control, and stochastic systems theory emphasizing the various connections between the different areas. The course assumes basic familiarity with ordinary differential equations and linear algebra. Main topics are

• vector fields on manifolds and their Lie algebras
• basic concepts of dynamical systems
• geometric control theory - accessibility and controllability
• control theory and Lie theory
• stochastic differential equations
• support theorems
• qualitative theory of stochastic systems

Fritz Colonius has added a chapter on
• optimal control theory

The goal of this course is to introduce students to the value of Hilbert space ideas for gaining geometric insight into a variety of problem areas addressed in statistics and random processes. The first half of the course covers some of the mathematics of Hilbert spaces. It begins with a review of linear vector spaces and operators, norms, inner products, completeness, linear functionals, and dual spaces. This is followed by the projection theorem, orthogonalization, quadratic forms, the Levinson system, optimization and Lagrange multipliers. The second half of the course applies these tools to problems in four areas of statistics: linear models, multivariate analysis, time series modeling, and filtering/prediction theory.

Reference material:
Finite Dimensional Vector Spaces, P. R. Halmos
Introduction to Hilbert Space, Halmos
Optimization by Vector Space Methods, Luenberger
Stochastic Processes, Frazho (personal notes)