The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the geometric optimization control of mechanical systems, such as robots and drones. I will also describe the role of geometry and geometric structure-preservation in accelerated optimization and adjoint sensitivity analysis, and how they can be used to train neural networks derived from neural differential equations.