Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

• Published in: Set-valued and variational analysis Vol. 27; no. 2; pp. 405 - 431
• Main Authors: Martens, Björn, Gerdts, Matthias
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 15.06.2019; Springer Nature B.V
• Subjects: Algebra; Analysis; Coercivity; Control theory; Convergence; Differential equations; Discretization; Economic models; Indexing; Mathematical analysis; Mathematics; Mathematics and Statistics; Optimal control; Optimization; Riccati-equation; Optimal control; Euler-discretization; Second order sufficient conditions; 65L80; 65L20; Differential-algebraic equation; 49K15; 49J15; Convergence analysis; Legendre-Clebsch
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-018-0471-x

For optimal control problems subject to index-one differential-algebraic equations in semi-explicit form we discuss second order sufficient conditions in form of a coercivity condition taking into account the two-norm discrepancy. Furthermore we introduce a related Riccati-type and Legendre-Clebsch condition which are sufficient for the validity of the coercivity condition. Using the implicit Euler-discretization we approximate the optimal control problem and analyze the convergence of solutions of the local minimum principle for the discretized optimal control problem by applying the general convergence framework of Stetter, which requires the discretization method to be continuous, consistent, and stable.