Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

• Published in: Computational optimization and applications Vol. 86; no. 3; pp. 1299 - 1325
• Main Author: Martens, Björn
• Format: Journal Article
• Language: English
• Published: New York, NY Springer US 01.12.2023; Springer Nature B.V
• Subjects: Algebra; Approximation; Convergence analysis; Convex and Discrete Geometry; Differential equations; Differential–algebraic equation; Discrete approximations; Errors; Estimates; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimal control; Optimization; Ordinary differential equations; Runge-Kutta method; Runge–Kutta schemes; Statistics; 65L06; Discrete approximations; Optimal control; Runge–Kutta schemes; 49M25; 49K15; Differential–algebraic equation; 49J15; 34A09; Convergence analysis
• ISSN: 1573-2894; 0926-6003; 1573-2894
• DOI: 10.1007/s10589-023-00484-1

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.