Metric Regularity Properties in Bang-Bang Type Linear-Quadratic Optimal Control Problems
- 30 Downloads
The paper investigates the Lipschitz/Hölder stability with respect to perturbations of optimal control problems with linear dynamic and cost functional which is quadratic in the state and linear in the control variable. The optimal control is assumed to be of bang-bang type and the problem to enjoy certain convexity properties. Conditions for bi-metric regularity and (Hölder) metric sub-regularity are established, involving only the order of the zeros of the associated switching function and smoothness of the data. These results provide a basis for the investigation of various approximation methods. They are utilized in this paper for the convergence analysis of a Newton-type method applied to optimal control problems which are affine with respect to the control.
KeywordsVariational analysis Optimal control Linear control systems Bang-bang controls Metric regularity Stability analysis Newton’s method
Mathematics Subject Classification (2010)49J30 49K40 49M15 49N05 47J07
Open access funding provided by Austrian Science Fund (FWF). This research is supported by the Austrian Science Foundation (FWF) under grant No P26640-N25. The second author is also supported by the Doctoral Program “Vienna Graduate School on Computational Optimization” funded by the Austrian Science Fund (FWF), project No W1260-N35.
- 7.Dontchev, A.L., Malanowski, K.: A characterization of Lipschitzian stability in optimal control. In: Calculus of Variations and Optimal Control, vol. 411, pp 62–76 (2000)Google Scholar
- 15.Haunschmied, J.L., Pietrus, A., Veliov, V.M.: The Euler method for linear control systems revisited. In: Proceedings of the 9th International Conference on Large-Scale Scientific Computing, pp 90–97 (2013)Google Scholar
- 16.Ioffe, A.D.: Metric Regularity. Theory and Applications - a survey. arXiv:1505.07920, 24 Oct 2015.
- 21.Maurer, H., Osmolovskii, N.P.: Applications to regular and bang-bang control: second-order necessary and sufficient optimality conditions in calculus of variations and optimal control. SIAM Adv. Des. Control 24 (2012)Google Scholar
- 25.Schneider, C., Wachsmuth, G.: Regularization and discretization error estimates for optimal control of ODEs with group sparsity. ESAIM: Control, Optimisation and Calculus of Variations (2017). https://doi.org/10.1051/cocv/2017049
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.