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It is a matter of observation that many complex and important systems evolve, and that this evolution depends, to an extent, on external stimuli (which we call controls). These stimuli are frequently varying with time, and on normal scales are often highly oscillatory and potentially non-differentiable. To model them mathematically one has to go beyond the classical theory of differential equations and find a meaning for
dyt⁄dt = ∑i f(yt) dyti⁄dt, y0 = a.
when γ is a rough path. The theory of rough paths draws together earlier perspectives and results of L. C. Young and K.T. Chen; it develops the analysis required to model these interactions without imposing the requirement that the control be differentiable.