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：本文提出了一种新的离散化方法，该方法在定义域两端同时进行配点，并能够实现与哈密顿边值问题相关联的协态变量的完全收敛。这一目标通过向勒让德-洛巴托多项式的根中引入一个*额外样本点*来实现，从而确保相应的微分矩阵具有满秩性质。我们研究了新样本点的位置，以使微分矩阵对扰动具有最高的鲁棒性，并证明该位置同时也是能够减轻多项式插值中龙格现象的最佳选择。本文通过两个基准问题的成功实现，验证了理论发现。观察表明，新方法随所用离散点数量呈指数级收敛。