Abstract
The paper presents a class of time-discretization schemes for terminal optimal control problems for linear systems. An error estimate is obtained for the optimal control and for the optimal performance, although the optimal control is typically discontinuous, and neither Lipschitz nor structurally stable with respect to perturbations.
This research was partially supported by the Austrian Science Found P18161.
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
Agrachev, A.A., Stefani, G., Zezza, P.: Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 41(4), 991–1014 (2002)
Doitchinov, B.D., Veliov, V.M.: Parametrisations of integrals of set-valued mappings and applications. J. of Math. Anal. and Appl. 179(2), 483–499 (1993)
Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numerical Anal. 38(1), 202–226 (2000)
Felgenhauer, U.: On stability of bang-bang type controls. SIAM J. Control Optim. 41(6), 1843–1867 (2003)
Felgenhauer, U.: On the optimality of optimal bang-bang controls for linear and semilinear systems. Control & Cybernetics (to appear)
Ferretti, R.: High-order approximations of linear control systems via Runge-Kutta schemes. Computing 58, 351–364 (1997)
Kostyukova, O., Kostina, E.: Analysis of properties of the solutions to parametric time-optimal problems. Computational Optimization and Applications 26, 285–326 (2003)
Maurer, H., Osmolovskii, N.: Second order sufficient conditions for time-optimal bang-bang control. SIAM J. Control Optim. 42(6), 2239–2263 (2004)
Noble, J., Schättler, H.: Sufficient conditions for relative minima of broken extremals in optimal control theory. J. Math. Anal. Appl. 269(1), 98–128 (2002)
Osmolovskii, N.P.: Second-order conditions for broken extremal. In: Calculus of variations and optimal control (Haifa, 1998), vol. 411, pp. 198–216. Chapman & Hall/CRC Res. Notes Math., Boca Raton (2000)
Pliś, A.: Accessible sets in control theory. In: International Conference on Differential Equations (Calif., 1974), pp. 646–650. Academic Press, London (1975)
Polovinkin, E.: Strongly convex analysis. Mat. Sb. 187(2), 103–130 (1996); translation in Sb. Math. 187(2), 259–286 (1996)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes. John Wiley & Sons, Chichester (1962)
Schwartz, A., Polak, E.: Consistent approximations for optimal control problems based on Runge-Kutta integration. SIAM J. Control Optim. 34, 1235–1269 (1996)
Veliov, V.M.: On the convexity of integrals of multivalued mappings: applications in control theory. J. Optim. Theory Appl. 54(3), 541–563 (1987)
Veliov, V.M.: Approximations to Differential Inclusions by Discrete Inclusions. In: WP–89–017, IIASA, Laxenburg, Austria (1989)
Veliov, V.M.: Error analysis of discrete approximations to bang-bang optimal control problems: the linear case. Control & Cybernetics (to appear)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Veliov, V.M. (2006). Approximations with Error Estimates for Optimal Control Problems for Linear Systems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2005. Lecture Notes in Computer Science, vol 3743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11666806_29
Download citation
DOI: https://doi.org/10.1007/11666806_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31994-8
Online ISBN: 978-3-540-31995-5
eBook Packages: Computer ScienceComputer Science (R0)