Riemannian optimal control

AbstractThe aim of this paper is to adapt the general multitime maxi-mum principle to a Riemannian setting. More precisely, we intend tostudy geometric optimal control problems constrained by the metriccompatibility evolution PDE system; the evolution (”multitime”) vari-ables are the local coordinates on a Riemannian manifold, the statevariable is a Riemannian structure and the control is a linear connec-tion compatible to the Riemannian metric. We apply the obtainedresults in order to solve two flow-type optimal control problems onRiemannian setting: firstly, we maximize the total divergence of afixed vector field; secondly, we optimize the total Laplacian (the gra-dient flux) of a fixed differentiable function. Each time, the result isa bang-bang-type optimal linear connection. Moreover, we emphasizethe possibility of choosing at least two soliton-type optimal (semi-)Riemannian structures. Finally, these theoretical examples help us toconclude about the geometric optimal shape of pipes, induced by thedirection of the flow passing through them.Keywords: multitime maximum principle, Riemannian optimalcontrol, shapeoptimization, gradient flow, total divergence, total Lapla-cian, bang-bang-type optimal solution, soliton-type metric.MSC2010: 49J20, 49N05, 49Q10, 53C05, 53C80.