In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately, their result is only concerned with fixed sampling times, and thus, it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely, we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally, a linear–quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.

在Bourdin和Trélat的最新论文中，已经提出了Pontryagin最大原理的一种形式（简称PMP），用于一般的非线性有限维最佳采样数据控制问题。不幸的是，它们的结果仅与固定采样时间有关，因此，它没有考虑自由采样时间的可能性。本文旨在填补文献中的空白。准确地说，我们建立了可以处理免费采样时间的PMP的新版本。就像在Bourdin和Trélat的上述工作中一样，我们获得了一阶必要的最优性条件，该条件被写为非正平均哈密顿梯度条件。此外，由于可以自由选择采样时间，我们得到一个新的和额外的必要最优条件，该条件恰好与哈密顿函数的连续性相吻合。在自主环境中，甚至可以导出哈密顿函数的恒定性。我们的证明基于Ekeland变分原理。最后，使用射击方法对一个线性二次方程进行了数值求解，说明了在固定采样时间情况下哈密顿函数的可能不连续性，并强调了在最佳采样时间情况下其连续性。