Abstract: 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. Rogers2007] from discrete to continuous time. We also make the link to old work of [Davis–Burstein,1992].