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

Convergence in nonlinear optimal sampled-data control problems