Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs
针对指标1微分代数方程的最优控制问题，提出一种将代数变量视为控制的新型龙格-库塔格式，推导出状态、控制及乘子的三阶误差估计。
Abstract In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.