Pontryagin maximum principle for optimal sampled-data control problems

TL;DR

This paper extends the Pontryagin maximum principle to optimal sampled-data control problems, discussing the maximization condition and demonstrating its application through a linear-quadratic parking problem example.

Contribution

It presents a version of the Pontryagin maximum principle tailored for sampled-data control problems and analyzes the maximization condition as an average of the classical weak maximum condition.

Findings

Derived a maximum principle for sampled-data controls

Analyzed the maximization condition as an average of classical conditions

Solved a linear-quadratic parking problem example

Abstract

In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. This result was recently obtained in [L. Bourdin and E. Tr{\'e}lat , Optimal sampled-data control, and generalizations on time scales, arXiv:1501.07361, 2015]. Then we discuss the maximization condition for optimal sampled-data controls that can be seen as an average of the weak maximization condition stated in the classical Pontryagin maximum principle for optimal (permanent) controls. Finally, applying this theorem, we solve a linear-quadratic example based on the classical parking problem.