Convergence of the Lagrange-Newton Method for Optimal Control Problems

• Autorzy: Malanowski K. D.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

optimal control; nonlinear ODEs; mixed constraints; Lagrange-Newton method

PL

sterowanie optymalne; ograniczenia mieszane; metoda Lagrange'a-Newtona

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2004
• Tom: Vol. 14, no 4
• Strony: 531--540
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Malanowski K. D.

• Systems Research Institute, Polish Academy of Sciences ul. Newelska 6, 01-447 Warszawa, Poland,

Bibliografia

• [1] 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.
• [2] Alt W. (1990a): Lagrange-Newton method for infinite-dimensional optimization problems. — Numer. Funct. Anal. Optim., Vol. 11, No. 3/4, pp. 201–224.
• [3] Alt W. (1990b): Parametric programming with applications to optimal control and sequential quadratic programming.— Bayreuther Math. Schriften, Vol. 34, No. 1, pp. 1–37.
• [4] Alt W. (1990c): Stability of solutions and the Lagrange-Newton method for nonlinear optimization and optimal control problems. — (Habilitationsschrift), Universität Bayreuth, Bayreuth.
• [5] 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.
• [6] 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.
• [7] Bonnans J.F. and Shapiro A. (2000): Perturbation Analysis of Optimization Problem. — New York: Springer.
• [8] Bulirsch R. (1971): Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. — Report of the Carl-Cranz-Gesellschaft, Oberpfaffenhofen, 1971.
• [9] 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.
• [10] Felgenhauer U. (2002): On stability of bang-bang type controls. — SIAM J. Contr. Optim., Vol. 41, No. 6, pp. 1843–1867.
• [11] 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.
• [12] Malanowski K. (1994): Regularity of solutions in stability analysis of optimization and optimal control problems.—Contr. Cybern., Vol. 23, No. 1/2, pp. 61–86.
• [13] Malanowski K. (1995): Stability and sensitivity of solutions to nonlinear optimal control problems. — Appl. Math. Optim., Vol. 32, No. 2, pp. 111–141.
• [14] Malanowski K. (2001): Stability and sensitivity analysis for optimal control problems with control-state constraints. — Dissertationes Mathematicae, Vol. CCCXCIV, pp. 1–51.
• [15] 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.
• [16] 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.
• [17] 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.
• [18] 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.
• [19] Maurer H. and Osmolovskii N. (2004): Second order optimality conditions for bang-bang control problems. — Contr. Cybern., Vol. 32, No. 3. pp. 555–584.
• [20] 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.
• [21] Robinson S.M. (1980): Strongly regular generalized equations. —Math. Oper. Res., Vol. 5, No. 1, pp. 43–62.
• [22] Stoer J. and Bulirsch R. (1980): Introduction to Numerical Analysis.— New York: Springer.