Optimal control for the Paneitz obstacle problem

TL;DR

This paper investigates an optimal control problem linked to the Paneitz obstacle problem on 4D manifolds, establishing existence, regularity, and characterization of optimal controls, especially on the standard sphere.

Contribution

It introduces a new optimal control framework for the Paneitz obstacle problem, proving existence, regularity, and explicit characterization of solutions.

Findings

Existence of optimal controls and states

Smoothness of optimal controls (C^{ abla}^)

Complete characterization on the 4D sphere

Abstract

In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C^{\infty}-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls.