Lectures: Friday 11-13 und Exercise Monday 11-12

The first event is Friday, 16 April. (The exercise of Monday, 12. April is canceled)

The lecture will be held over Zoom . The exercise will be a mix of discussions and questions. There will be sporadic exercise sheets as brain teasers, but no correction or grading.

The course exercises, skript etc. can be found in Heibox.

Contact: Lucas Dahinden, ldahinden-at-mathi.uni-heidelberg.de

Requirements: Any student with a Bachelor in Mathematics or Physics should be well prepared. An interest in geometry, dynamics or variational principles certainly helps.

We spice up the study of ODE's by adding a steering wheel. Instead of just following a vector field, we are able to correct the vector field at any time (+technical details).

Obvious real life examples for this setup are cars and spaceships, which (unsteered) follow a trajectory determined by physics until the pilot decides to turn the wheel. (Possibly) not so obvious examples include moduli spaces in thermodynamics (the steering wheel is an engine) or the stockmarket (the steering wheel is the decision to buy or sell stocks).

What to do with this new gained freedom? Optimise something, of course! In the examples above the quantity to optimise is expended fuel or arrival time (spaceship), energy lost to entropy increase (thermodynamics) or money gained (stockmarket). Of course, one may also try to optimise any other quantity.

We concentrate on the theoretic development of the theory. To this end, we first study controllability in linear systems, then pass on to fiberwise linear systems. Finally we turn to the theoretic main goal, the Pontrjagin maximum principle. If time permits, we delve deeper into topics around Sub-Riemannian geometry, which is a type of geometry that naturally arises in fiberwise linear systems.