Convergence in nonlinear optimal sampled-data control problems

TL;DR

This paper proves that, under certain conditions, the optimal solutions and costates of sampled-data control problems converge to those of the continuous-time problem as the sampling becomes finer, establishing a link between discrete and continuous optimal control.

Contribution

It demonstrates the convergence of optimal states and costates in nonlinear sampled-data control problems to their continuous counterparts as sampling intervals shrink, under specific assumptions.

Findings

Optimal state convergence as partition norm tends to zero

Costate convergence under nondegeneracy conditions

Control sampling commutes with Pontryagin maximum principle in the limit

Abstract

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.