Course detail

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.


Not applicable.

Recommended optional programme components

Not applicable.

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

Czech, English

Work placements

Not applicable.


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

Classification of course in study plans

  • Programme D-APM-K Doctoral, 1. year of study, winter semester, 0 credits, recommended

  • Programme D4P-P Doctoral

    branch D-APM , 1. year of study, winter semester, 0 credits, recommended

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.