We consider the binary supervised classification problem with the Gaussian functional model introduced in [7]. Taking advantage of the Gaussian structure, we design a natural plug-in classifier and derive a family of upper bounds on its worst-case excess risk over Sobolev spaces. These bounds are parametrized by a separation distance quantifying the difficulty of the problem, and are proved to be optimal (up to logarithmic factors) through matching minimax lower bounds. Using the recent works of [9] and [14] we also derive a logarithmic lower bound showing that the popular k-nearest neighbors classifier is far from optimality in this specific functional setting.Theorem (A). The classifier Φ dn defined in (16) with dn ≈ n 1 2s+1 has an excess risk roughly bounded by (omitting logarithmic factors and constant factors depending only on s and R): for n NR,s large enough,• In Section 4.1 we derive a matching minimax lower bound (up to logarithmic factors) showing that the above worst-case bound cannot be improved by any classifier.Theorem (B). For any number n NR,s of observations, any classifier Φ must satisfy (omitting again logarithmic factors and constant factors depending only on s and R):As an example, if we assume that we observe dX(t) = f (t)dt with X(0) = 0, then η(X) = exp( 1 2 f −g 2 ) 1+exp( 1 2 f −g 2 )