Geometric Control Theory (FSI-9GTR)

Academic year 2020/2021
Supervisor: doc. Mgr. Jaroslav Hrdina, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
Building the basics of geometric control theory. Ability to apply theory to engineering problems.
Learning outcomes and competences:
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.
Prerequisites:
The knowledge of mathematics gained within the bachelor's study programme.
Course contents:
Advanced Differential Geometry and Representation Theory in the theory Optimal Transport of Non-Holonomic Systems. Algebraic view of the dynamic systems.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Výuka se odehrává formou přednášky a není kontrolovaná
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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.
Literature - fundamental:
1. Y.L. Sachkov. Control theory on lie groups. J Math Sci, 156(3):381--439, 2009.
2. L. Zexiang, S. Sastry , R. M. Murray, A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.
3. Enrico Le Donne, Lecture notes on sub-Riemannian geometry, University of Jyväskylä
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D4P-P full-time study D-APM Applied Mathematics -- DrEx 0 Recommended course 3 1 W
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 W