Application of sub-Riemannian geometry in optimal control theory (Spring 2021)

We have recently adopted a method for finding such optimal trajectories which goes under the name of symmetry reduction method. On our space we consider the action of a transformation group which maps trajectories into trajectories without modifying the cost. This way we can consider a quotient space which is smaller than the original space and treat the optimization problem on this smaller space. Using this method we have successfully derived the form of the sub-Riemannian geodesics for several problems in quantum and classical mechanics. The success of this method rises the question of whether it can always be applied once we have a system with nonholonomic constraints in small dimensions.

The goal of this project is to investigate such a question by generating and studying simple examples in R^3. The activity will proceed by first describing some background material in Riemannian and sub-Riemannian geometry and then discussing some of the examples that have been already treated. Then the students will be encouraged to provide their own examples and discuss the feasibility of the technique in each case.

For more information contact Zhifei Zhu (zhifeiz@iastate.edu)

People:

• Zhifei Zhu (Postdoc)

• Domenico D’Alessandro (Faculty)

Pre-requisites:

• Multivariable calculus.

• Linear algebra.

• Ordinary differential equation.

• Experience with proofs.

• Topology and Real analysis is desirable, but not necessary.