Course detail
Geometric Control Theory
FSI-9GTR Acad. year: 2024/2025 Winter semester
Advanced Differential Geometry and Representation Theory in the theory Optimal Transport of Non-Holonomic Systems. Algebraic view of the dynamic systems.
Language of instruction
Czech
Supervisor
Department
Entry knowledge
The knowledge of mathematics gained within the bachelor's study programme.
Rules for evaluation and completion of the course
The course is finished by written and oral examination. The written part is 80% and the oral part 20% of the grade.
Výuka se odehrává formou přednášky a není kontrolovaná
Aims
Building the basics of geometric control theory. Ability to apply theory to engineering problems.
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 study programmes with the given course
Programme D-APM-K: Applied Mathematics, Doctoral, recommended course
Programme D-APM-P: Applied Mathematics, Doctoral, recommended course
Type of course unit
Lecture
20 hours, optionally
Syllabus
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.