This paper introduces a new method of discretization that collocates both endpoints of the domain and enables the complete convergence of the costate variables associated with the Hamilton boundary-value problem. This is achieved through the inclusion of an \emph{exceptional sample} to the roots of the Legendre-Lobatto polynomial, thus promoting the associated differentiation matrix to be full-rank. We study the location of the new sample such that the differentiation matrix is the most robust to perturbations and we prove that this location is also the choice that mitigates the Runge phenomenon associated with polynomial interpolation. Two benchmark problems are successfully implemented in support of our theoretical findings. The new method is observed to converge exponentially with the number of discretization points used.

翻译：本文提出一种新的离散化方法，该方法对定义域的两个端点均进行配点，并实现了与哈密顿边值问题相关联的协态变量的完全收敛。这是通过在勒让德-洛巴托多项式的根中引入一个"异常样本"来实现的，从而使得相关微分矩阵达到满秩。我们研究了新样本的位置，以使微分矩阵对扰动具有最强的鲁棒性，并证明该位置同时也是减轻多项式插值中龙格现象的最佳选择。两个基准问题的成功实施验证了我们的理论发现。观察到，新方法随着所用离散点数量的增加呈指数收敛。