Convergence in Nonlinear Optimal Sampled-Data Control Problems

Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution $x^*$. 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 partition tends to zero. Moreover, applying the Pontryagin maximum principle (PMP) 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 normality assumption, control sampling commutes, at the limit of small partitions, with the application of the PMP.