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): M 2:00pm-3:00pm, W 09:30am-10:30 am, F  10:00 am-11:00 am or by appointment.
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko
Home-page of the course: /~zelenko/F13M666.html

Class hours: MWF 11:30-12:20 BLOC 624

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.

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.

Course Description.

The course will mainly consist of 2 parts:  Controllability and Optimal Control.

Part 1. Controllability

a. Vector fields, flows, commutators of flows, Lie brackets of vector fields;

b. Orbits and attainable sets of families of vector fields.

c. Controllability of linear system (the Kalman rank condition). Local and global linearizability criteria for affine control systems.

d. Bracket generating families of vector fields (the Rashevsky-Chow theorem) with application to car-like robots (car with trailers systems).

e. The notion of manifolds and immersed submanifold. Nagano-Sussmann orbit theorem (discussion and sketch of the proof). Lie determined systems (including
real-analytic systems). Involutive distributions and Frobenius theorem. Application: analysis of the orbits for control of rigid body by jet torques.

f. The structure of attainable sets for Lie determined systems (Krener's theorem). Various saturation technique for deciding controllability:
convexification and conification, Poisson stable vector fields. Applications: Attainable sets for control of rigid body by jet torques

g. Some elements of the theory of (matrix) Lie groups. Controllability of left invariant affine control systems on compact Lie groups. Application: control
of finite level quantum systems, control of Frenet frame of curves in Euclidean space by (higher degree) curvature.

Part 2 Optimal Control

a. Statement of various optimal control problem: with fixed and free terminal time, time-optimal problems, with free boundary conditions, Mayer's problem.

b. Reduction of the optimal control problem to study of the attainable sets. Pontryagin Maximum Principle: Formulation and proof. Examples: fastest stop of
the train; control of linear oscillator (without and with friction), Dubins car.

c. Elements of the theory of linear time-minimal problems: General Position Condition, Theorem on bang-bang extremals, Theorems on existence and uniqueness
of optimal control, Condition for uniqueness of extremal control, estimates for number of switches. Various examples of optimal synthesis of linear
time-optimal control systems on the plane with one and two inputs.

d. Elements of the theory of linear-quadratic problems: extremals, the notion of conjugate points of linear-quadratic problems and its role in the existence
and uniqueness of optimal control of linear-quadratic problems.

e. Elements of symplectic geometry: introduction to theory of exterior differential forms; cotangent bundle, canonical symplectic structure on cotangent
bundle, Hamiltonian vector fields, Lagrangian submanifolds, Poincare-Cartan integral invariant.

g. Elements of theory of fields of extremals: Legendre condition, Jacobi equation and conjugate points along normal Pontryagin extremals, Theorem on
optimality in absence of conjugate points.


Scholastic dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules .

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.


Americans with Disabilities Act (ADA) Policy Statement. The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, in Cain Hall, Room B118, or call 845-1637. For additional information visit http://disability.tamu.edu.