I will introduce the notion of "limit solution" for nonlinear o.d.e.'s where a control u is present together its time derivative. The control u is allowed to be a discontinuous integrable function,so that its derivative gives the system an impulsive character. Limit solutions subsume the main former notions of solution for systems with "generalized" controls. Successively I will address the question whether or not a minimum problem connected with limit solutions is a proper extension of the minimum problem associated with regular solutions. We shall see that in general the answer is negative: this means that a gap can occur between the infimum values of the two problems. Finally, I will present a geometrical condition, here called "quick reachability" which proves to be sufficient to avoid such gaps.