Convergence results for two Lagrange-Newton-type methods of solving optimal control problems are presented. It is shown how the methods can be applied to a class of optimal control problems for nonlinear ODEs, subject to mixed control-state constraints. The first method reduces to an SQP algorithm. It does not require any information on the structure of the optimal solution. The other one is the shooting method, where information on the structure of the optimal solution is exploited. In each case, conditions for well-posedness and local quadratic convergence are given. The scope of applicability is briefly discussed.
@article{bwmeta1.element.bwnjournal-article-amcv14i4p531bwm,
author = {Malanowski, Kazimierz},
title = {Convergence of the Lagrange-Newton method for optimal control problems},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {14},
year = {2004},
pages = {531-540},
zbl = {1082.49033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv14i4p531bwm}
}
Malanowski, Kazimierz. Convergence of the Lagrange-Newton method for optimal control problems. International Journal of Applied Mathematics and Computer Science, Tome 14 (2004) pp. 531-540. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv14i4p531bwm/
[000] Agrachev A.A., Stefani G. and Zezza P.L. (2002): Strong optimality for a bang-bang trajectory. -SIAM J. Contr. Optim., Vol. 41, No. 4, pp. 991-1014. | Zbl 1020.49021
[001] Alt W. (1990a): Lagrange-Newton method for infinite-dimensional optimization problems. -Numer. Funct. Anal. Optim., Vol. 11, No. 34, pp. 201-224. | Zbl 0694.49022
[002] Alt W. (1990b): Parametric programming with applications to optimal control and sequential quadratic programming. - Bayreuther Math. Schriften, Vol. 34, No. 1, pp. 1-37. | Zbl 0734.90094
[003] Alt W. (1990c): Stability of solutions and the Lagrange-Newton method for nonlinear optimization andoptimal control problems. - (Habilitationsschrift), Universitat Bayreuth, Bayreuth.
[004] Alt W. and Malanowski K. (1993): The Lagrange-Newton method for nonlinear optimal control problems. - Comput. Optim. Appl., Vol. 2, No. 1, pp. 77-100. | Zbl 0774.49022
[005] Alt W. and Malanowski K. (1995): The Lagrange-Newton method for state constrained optimal control problems. - Comput. Optim. Appl., Vol. 4, No. 3, pp. 217-239. | Zbl 0821.49024
[006] Bonnans J.F. and Shapiro A. (2000): Perturbation Analysis of Optimization Problem. - New York: Springer. | Zbl 0966.49001
[007] Bulirsch R. (1971): Die Mehrzielmethode zur numerischen Losung von nichtlinearen Randwert problemen und Aufgaben der optimalen Steuerung. - Report of the Carl-Cranz-Gesellschaft, Oberpfaffenhofen, 1971.
[008] Dontchev A.L. and Hager W.W. (1998): Lipschitz stability for state constrained nonlinear optimal control. - SIAM J. Contr. Optim., Vol. 35, No. 2, pp. 696-718. | Zbl 0917.49025
[009] Felgenhauer U. (2002): On stability of bang-bang type controls. - SIAM J. Contr. Optim., Vol. 41, No. 6, pp. 1843-1867. | Zbl 1031.49026
[010] Kim J.-H.R. and Maurer H. (2003): Sensitivity analysis of optimal control problems with bang-bang controls. -Proc. 42nd IEEE Conf. Decision and Control, CDC'2003, Maui, Hawaii, USA, pp. 3281-3286.
[011] Malanowski K. (1994): Regularity of solutions in stability analysis of optimization and optimal control problems. - Contr. Cybern., Vol. 23, No. 12, pp. 61-86. | Zbl 0810.49009
[012] Malanowski K. (1995): Stability and sensitivity of solutions to nonlinear optimal control problems. - Appl. Math. Optim., Vol. 32, No. 2, pp. 111-141. | Zbl 0842.49020
[013] Malanowski K. (2001): Stability and sensitivity analysis for optimal control problems with control-state constraints. - Dissertationes Mathematicae, Vol. CCCXCIV, pp. 1-51. | Zbl 1017.49027
[014] Malanowski K. and Maurer H. (1996a): Sensitivity analysis for parametric optimal control problems with control-state constraints. - Comput. Optim. Appl., Vol. 5, No. 3, pp. 253-283. | Zbl 0864.49020
[015] Malanowski K. and Maurer H. (1996b): Sensitivity analysis for state-constrained optimal control problems. - Discr. Cont. Dynam. Syst., Vol. 4, No. 2, pp. 241-272. | Zbl 0952.49022
[016] Malanowski K. and Maurer H. (2001): Sensitivity analysis for optimal control problems subject to higher order state constraints. - Ann. Oper. Res., Vol. 101, No. 2, pp. 43-73. | Zbl 1005.49021
[017] Maurer H. and Oberle J. (2002): Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. - SIAM J. Contr. Optim., Vol. 41, No. 2, pp. 380-403. | Zbl 1012.49018
[018] Maurer H. and Osmolovskii N. (2004): Second order optimality conditions for bang-bang control problems. - Contr. Cybern., Vol. 32, No. 3. pp. 555-584. | Zbl 1127.49019
[019] Maurer H. and Pesch H.J. (1994): Solution differentiability for parametric optimal control problems with control-state constraints. -Contr. Cybern., Vol. 23, No. 1, pp. 201-227. | Zbl 0809.93024
[020] Robinson S.M. (1980): Strongly regulargeneralized equations. - Math. Oper. Res., Vol. 5, No. 1, pp. 43-62. | Zbl 0437.90094
[021] Stoer J. and Bulirsch R. (1980): Introduction to Numerical Analysis. - New York: Springer. | Zbl 0423.65002