Syllabus for  Geometric Control Theory (formal title "Seminar in Geometry") Fall 2011 (Math 666),
Fall 11

Instructor: Igor Zelenko
Office: Milner 324
Office hours (subject to change): Mon 2:00 p.m.- 3:00 p.m., Wed 2:00 p.m.-3:00 p.m., Fri 2:00 p.m. -3:00 p.m., or by appointment.
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko

Class hours: MWF 11:30-12:20 ZACH119D

Text:.
A. A. Agrachev, Yu.L. Sachkov, Control theory from the geometric viewpoint, Springer Verlag, 2004
Note that this book requires a higher background than the prerequisites of the course, but I will make all efforts to adjust your background to the level of the book.

Course Description.

The course will mainly consist of 2 parts:

i. Controllability: orbits of families of vector fields (Nagano-Sussmann, Rashevsky-Chow, and Frobenius theorems), the structure of attainable sets for bracket-generating systems (accessibility, relaxation, Poisson stability) with application to control of rigid body and control of configurations. This corresponds to chapters 1, 2, 3, 5, 6, 7, and 8 of the book;

ii. Optimal Control Theory: existence theorems (Filippov theorem), Pontryagin Maximum Principle with applications, linear time-optimal problems, linear- quadratic problems, dynamic programming and Hamilton-Jacobi equation. Thi corresponds to chapters 10, 12, 13, 15, 16, 17. If the time permit we will discuss second order optimality conditions ( second variation and Jacobi fields , chapters 20 and 21)

If the time permit we also will discuss state-feedback equivalence of control systems: state and state-feedback linearizability (chapters 4 and 9)  and  Hamiltonian approach to Differential Geometry of control systems (chapter 23) (this part does not require any preliminary background in Differential Geometry).


Prerequisite:

The course does not have any graduate course as a prerequisite. The undergraduate prerequisites are standard Calculus courses (MATH 151, 152, 251 or equivalent), Differential Equations (MATH 308), and Linear algebra (MATH 304 or 323). I will give all mathematical background beyond the above courses in the class.

Grading. Your grade will be determined by  biweekly home assignments (70%)  and final take-home exam or project (30%).  The grade will be given according to the following percentage:

85%-100%=A, 75%-84%=B , 65%-74%=C, 55%-64%=D, less than 55%=F

Additional sources:
1. Regarding controllability: V. Jurdjevich, Geometric Control Theory, Cambridge Studies in Advanced Mathematics 52, 1997
2. Regarding Optimal Control: L. Pontryagin, V. Boltyansky, R. Gamkrelidze, and E. Mishchenko, The mathematical theory of optimal processes, Wiley-Interscience, New York, 1962.

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Copyright policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.


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