SISSA Functional Analyisis Sector
Course Announcement,  Spring 2002
 

Geometric Control Theory and Applications


Lecturer:  Claudio Altafini 
  altafini@sissa.it
  ~altafini

Schedule

first class: 9th April 2002 at 11:00 in room L

Format

9-10 lectures, (1h30 each)
 

Tentative Course Outline

The syllabus is subject to change according to audience background and/or interests.
 
Introductory material
  1. manifolds, vector fields, tangent spaces;
  2. orbits of vector fields and Frobenius Theorem;
  3. controllablity and Chow Theorem;
  4. drift versus driftless systems, accessibility versus controllability
Bilinear control systems
  1. bilinear systems and matrix transition Lie groups;
  2. structure of matrix Lie groups (homogeneous spaces, transitivity, exponential map and canonical coordinates) 
  3. Lie algebras (Levi decomposition, semisimpicity, solvability, nilpotency, Cartan criteria);
  4. controllability properties for bilinear control systems on matrix Lie groups;
  5. series expansions of control systems;
  6. nonholonomic behavior in robotic systems: examples systems with of first order nonholonomic constraints (trailer systems, chained form);
  7. examples of control systems on Lie groups (rigid bodies on SO(3) and SE(3); system on a sphere)
Open-loop control methods for bilinear systems
  1. piecewise-constant controls
  2. feedback linearization
  3. system inversion
  4. averaging and oscillatory control
Application of bilinear systems: finite dimensional Schrodinger equation
  1. matrix transition Lie group of a Schordinger equation: SU(N);
  2. controllability on semisimple Lie algebras;
  3. control by piecewise constant inputs;
  4. functional series expansions in the controls: Magnus expansion and product of exponentials expansions; 
  5. application to elementary unitary gates synthesis for "quantum circuits";
  6. example two-level system: qbit.