Controlled geometric flows of maps

In this talk, we discuss controlled geometric flows of maps, motivated by the heat flow and wave maps, in which control is incorporated into geometric evolution equations. Unlike classical control problems in flat PDE settings, the geometry and topology of the target manifold directly influence both the dynamics and its controllability, leading to phenomena that do not arise in Euclidean models. This perspective sheds light on small-time global controllability and provides qualitative characterizations, including connections between global controllability and homotopy.

Based on joint work with Jean-Michel Coron and Joachim Krieger.