  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.
