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. Trélat, 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.
Bourdin, L., & Trélat, E. (2015). Pontryagin maximum principle for optimal sampled-data control problems. In IFAC-PapersOnLine (Vol. 28, pp. 80–84). https://doi.org/10.1016/j.ifacol.2015.11.063