Geometric Control Theory
FSI-9GTRAcad. year: 2020/2021
Advanced Differential Geometry and Representation Theory in the theory Optimal Transport of Non-Holonomic Systems. Algebraic view of the dynamic systems.
Learning outcomes of the course unit
Students will learn to use advanced parts of differential geometry and representation theory. For a specific mechanism: the construction of kinematic chain, the solution of differential kinematics, design of optimal trajectory.
The knowledge of mathematics gained within the bachelor's study programme.
Recommended optional programme components
Recommended or required reading
Y.L. Sachkov. Control theory on lie groups. J Math Sci, 156(3):381--439, 2009. (EN)
L. Zexiang, S. Sastry , R. M. Murray, A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994. (EN)
Enrico Le Donne, Lecture notes on sub-Riemannian geometry, University of Jyväskylä (EN)
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline.
Assesment methods and criteria linked to learning outcomes
The course is finished by written and oral examination. The written part is 80% and the oral part 20% of the grade.
Language of instruction
Building the basics of geometric control theory. Ability to apply theory to engineering problems.
Specification of controlled education, way of implementation and compensation for absences
Výuka se odehrává formou přednášky a není kontrolovaná
Type of course unit
20 hours, optionally
Teacher / Lecturer
1. Lie algebras, definitions and basic concepts, examples (orthogonal, special, Heisenberg, etc. ), adjoint representation, semi-simple, solvable and nilpotent Lie algebras.
2. Algebra of controllability, configuration space, non-homonomous conditions, differential kinematics, Pffaf's system, vector fields and bracket.
3. Nilpotent approximations (symbols), definitions and basic properties, adapted and privileged coordinates, Bellaiche's Algorithm.
4. Lie groups. definitions, examples (special, orthogonal, spin, etc.), Lie algebra as the tangent space of Lie groups.
5. Leftinvariant vector fields, definition, Lie algebra of left-vector vector fields, flows of vector fields, a group structure under of nilpotent Lie algebras.
6. Sub - Riemanian (sR) geometry, distribution, sR-metric, horizontal curves.
7. Minimal curves (local extremals), PMP for nilpotent approximations, normal and abnormal extremals, sR-Hamiltonian
8. Heisenberg geometry, Heisenberg's group and algebra, description of the mechanism known as dubin car.
9. Other Structures on Heisenberg geometry. Overview of Heisenberg Geometry, Lagrange and CR Geometry. Infinitesimal automorphisms.
10. Conjunction points. Fixed points of infinitesimal automorphisms. Heisenberg's apple.