Riemannian optimal control
  The aim of this paper is to adapt the general multitime maximum principle to a Riemannian setting. More precisely, we intend to study geometric optimal control problems constrained by the metric compatibility evolution PDE system; the evolution ("multitime") variables are the local coordinates on a Riemannian manifold, the state variable is a Riemannian structure and the control is a linear connection compatible to the Riemannian metric. We apply the obtained results in order to solve two flow-type optimal control problems on Riemannian setting: firstly, we maximize the total divergence of a fixed vector field; secondly, we optimize the total Laplacian (the gradient flux) of a fixed differentiable function. Each time, the result is a bang-bang-type optimal linear connection. Moreover, we emphasize the possibility of choosing at least two soliton-type optimal (semi-) Riemannian structures. Finally, these theoretical examples help us to conclude about the geometric optimal shape of pipes, induced by the direction of the flow passing through them. 
Andreea Bejenaru and Constantin Udriste
math.OC math.DG