  We study a class of controlled rough differential equations. It is shown that
the value function satisfies a HJB type equation; we also establish a form of
the Pontryagin maximum principle. Deterministic problems of this type arise in
the duality theory for controlled diffusion processes and typically involve
anticipating stochastic analysis. We propose a formulation based on rough paths
and then obtain a generalization of Roger's duality formula [L. C. G. Rogers,
2007] from discrete to continuous time. We also make the link to old work of
[Davis--Burstein, 1987].
