  Consider, on the one part, a general nonlinear finite-dimensional optimal
control problem and assume that it has a unique solution whose state is denoted
by $x^*$. On the other part, consider the sampled-data control version of it.
Under appropriate assumptions, we prove that the optimal state of the
sampled-data problem converges uniformly to $x^*$ as the norm of the
corresponding partition tends to zero. Moreover, applying the Pontryagin
maximum principle to both problems, we prove that, if $x^*$ has a unique weak
extremal lift with a costate $p$ that is normal, then the costate of the
sampled-data problem converges uniformly to $p$. In other words, under a
nondegeneracy assumption, control sampling commutes, at the limit of small
partitions, with the application of the Pontryagin maximum principle.
