Abstract
Necessary optimality conditions are statements about nothing if an optimal solution does not exist. Therefore it makes sense to embed such conditions in a general framework which excludes the possibility of empty (even false) assertions all the more as the assumptions for existence results are much stronger than those for necessary optimality conditions. We call such a framework suboptimality theorem and exemplify it by a simple problem of optimal control. Furthermore, we investigate the set of suboptimal solutions by means of a new theorem about the approximation of measurable by simple functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Balder, E. J.: New Existence results for Optimal Controls in the Absence of Convexity: the Importance of Extremality, SLAM J. Control Optimization 32 (1994) No.3, 890–916.
Ekeland, I.: On the Variational Principle, J. Math. Anal. Appl. 47 (1974) 324–353.
Fattorini, H. O.: The Maximum Principle for Nonlinear Nonconvex Systems in Infinite Dimensional Spaces, in: Lecture Notes in Control and Information Science 75 (Springer-Verlag New York 1985) 162–178.
Fattorini, H. O.; Frankowska, H.: Necessary Conditions for Infinite-Dimensional Control Problems, Math. Control Signals Systems 4 (1991) 41–67.
Hamel, A.: Anwendungen des Variationsprinzips von Ekeland in der Optimalen Steuerung, Doct. Diss., Halle/Saale 1996.
Hamel, A.: Suboptimal Solutions of Control Problems for Distributed Parameter Systems, to appear in Proceedings of the 8th French German Conference on Optimization, LN in Economics and Math. Systems, Springer-Verlag 1997.
Ioffe, A. D.; Tichomirov, V. M.: Theorie der Extremalaufgaben, Deutscher Verlag der Wissenschaften, Berlin 1979.
Klötzler. R.: Optimal Transportation Flows, ZAA 14 (1995) No. 2, 391–401.
Penot, J. P.: The Drop Theorem, the Petal Theorem and Ekeland’s Variational Principle, Nonlinear Analysis, Theory, Methods & Appl. 10 (1986) No. 9, 813–822.
Plotnikov, V. I.; Sumin, M. I.: The Construction of Minimizing Sequences in Problems of the Control of Systems with Distributed Parameters, U.S.S.R. J. Comput. Math. Math. Phy. 22, No. 1 (1982) 49–57.
Taylor, S. J.: Introduction to Measure and Integration, Cambrigde University Press 1966.
Young, L. C.: Calculus of Variations and Optimal Control Theory, Chelsea Publishing Company New York 1980.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this paper
Cite this paper
Hamel, A. (1998). Suboptimality Theorems in Optimal Control. In: Schmidt, W.H., Heier, K., Bittner, L., Bulirsch, R. (eds) Variational Calculus, Optimal Control and Applications. International Series of Numerical Mathematics, vol 124. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8802-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8802-8_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9780-8
Online ISBN: 978-3-0348-8802-8
eBook Packages: Springer Book Archive