ETD

• energy-decreasing flow
• Lie groups
• sub-Riemannian geometry

The main objective of this thesis is to define a curve flow that reduces the energy preserving the endpoints. Following some work of Capogna and Le Donne, we will define a sort of gradient flow on the set of controls with the same endpoint. We will call the obtained flow the Control Flow. We will then proceed to prove a result of existence and uniqueness outside the set of singular controls and highlight some issues regarding the singular controls. Lastly, we will present some simple examples.