Abstract: In a controlled differential equation, the solution depends on the driving path only through its iterated integrals. We are interested in whether the iterated integrals of the paths essentially determine the paths. B. Hambly and T. Lyons gave a positive answer for paths with bounded variation in 2006, but the analogous problem for rough paths has been open since then. We propose a proof for this conjecture for a class of rough paths. Joint work with X. Geng, T. Lyons and D. Yang.