Convergence Analysis for Approximations of Optimal Control Problems Subject to Higher Index Differential-Algebraic Equations and Pure State Constraints

Martens, Björn; Gerdts, Matthias

doi:10.1137/20m1353952

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Cited by 2 publications

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“…is Lipschitz continuous with respect to (X , Y ) close to ( X (t), Ŷ (t)) for some t ∈ [0, 1]. Using the sensitivities ( 27), ( 28), (30), (31) we get…”

Section: Proofmentioning

confidence: 99%

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

Martens

2023

Comput Optim Appl

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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.

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“…is Lipschitz continuous with respect to (X , Y ) close to ( X (t), Ŷ (t)) for some t ∈ [0, 1]. Using the sensitivities ( 27), ( 28), (30), (31) we get…”

Section: Proofmentioning

confidence: 99%