Riemannian optimal control

TL;DR

This paper extends the multitime maximum principle to Riemannian geometry, solving flow-type optimal control problems that yield bang-bang solutions and insights into optimal geometric structures like pipes.

Contribution

It introduces a Riemannian adaptation of the multitime maximum principle and applies it to optimize geometric control problems involving divergence and Laplacian.

Findings

Optimal linear connections are bang-bang type.

Existence of soliton-type optimal Riemannian structures.

Insights into the geometric shape of flow-induced pipes.

Abstract

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.