A smooth optimal control problem with mixed constraints is considered. Under the normality assumption, a proof of second-order necessary optimality conditions based on the Robinson stability theorem is provided. The main feature of the obtained result is that the local regularity with respect to the mixed constraints is assumed, that is, a regularity in an e -tube along the minimizer, but not the conventional global regularity hypothesis. This impacts the maximum condition. Therefore, the normal set of Lagrange multipliers fulfills the LegendreClebsch condition and the maximum principle. At the same time, the maximum condition is modified since, now, the maximum is taken over a reduced feasible set. Furthermore, the case of abnormal minimizers is considered. The same type of reduced maximum condition is obtained along with a refined Legendre-Clebsch condition which is meaningful in the abnormal case.