SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 1903-1926, January 2021.
In this paper, we derive error estimates for implicit Euler discretizations of optimal control problems subject to index two differential-algebraic equations and first-order pure state constraints. First, we deduce discrete necessary conditions, which are consistent with the continuous necessary conditions, by transforming the discrete Lagrange multipliers. Then, we prove that, for sufficiently small perturbations, a perturbed version of the discretized problem has a solution, which satisfies an error estimate depending on the perturbation and mesh size with respect to the continuous solution. We apply this result to a subclass of perturbed problems and further prove that the solution of this subclass is Lipschitz continuous with respect to the perturbation and constant of order $O(h)$. This implies a linear convergence rate for the discrete states, control, and multipliers in the $L_\infty$-norm.

中文翻译：

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高指数微分-代数方程和纯状态约束下最优控制问题逼近的收敛性分析

SIAM控制与优化杂志，第59卷，第3期，第1903-1926页，2021年1月。
在本文中，我们推导了针对带有索引两个微分代数方程和一阶纯状态约束的最优控制问题的隐式欧拉离散化的误差估计。首先，我们通过变换离散拉格朗日乘数来推导与连续必要条件一致的离散必要条件。然后，我们证明，对于足够小的扰动，离散问题的扰动版本具有解，该解满足依赖于相对于连续解的扰动和网格大小的误差估计。我们将此结果应用于扰动问题的子类，并进一步证明该子类的解关于Lipschitz关于扰动和阶次$ O（h）$的常数是连续的。