We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of Lyons. 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 make the link to old work of Davis and Burstein (Stoch Stoch Rep 40: 203256, 1992) and then prove a continuous-time generalization of Roger's duality formula [SIAM J Control Optim 46: 1116-1132, 2007]. The generic case of controlled volatility is seen to give trivial duality bounds, and explains the focus in Burstein-Davis' (and this) work on controlled drift. Our study of controlled rough differential equations also relates to work of Mazliak and Nourdin (Stoch Dyn 08: 23, 2008).