Optimization methods for the verification of second order sufficient conditions for bang-bang controls

Optimal Control Applications and Methods

Volumn 26, Issue 3, 2005, Pages 129-156

  Maurer, H ^a   Buskens C ^b   Kim, J H R ^a   Kaya, C Y ^c  

^a UNIVERSITY OF MÜNSTER   (Germany)

^b UNIVERSITY OF BREMEN   (Germany)

^c UNIVERSITY OF SOUTH AUSTRALIA   (Australia)

Author keywords

Bang bang control; Control of a batch reactor; Control of a van der Pol oscillator; Non linear programming; Robot control; Second order sufficient conditions

Indexed keywords

COMPUTATIONAL METHODS; COMPUTER SOFTWARE; LYAPUNOV METHODS; NONLINEAR PROGRAMMING; NUMERICAL ANALYSIS; OPTIMIZATION; ROBOTS;

BANG-BANG CONTROL; CONTROL OF A BATCH REACTORS; CONTROL OF A VAN DER POL OSCILLATOR; ROBOT CONTROL; SECOND ORDER SUFFICIENT CONDITIONS;

OPTIMAL CONTROL SYSTEMS;

EID: 21244449627     PISSN: 01432087     EISSN: None     Source Type: Journal    
DOI: 10.1002/oca.756     Document Type: Article

References (36)

1. Malanowski K, Maurer H. Sensitivity analysis for parametric control problems with control-state constraints. Computational Optimization and Applications 1996; 5:253-283.
2. Malanowski K, Maurer H. Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems 1998; 4:241-272.
3. Maurer H, Augustin D. Sensitivity analysis and real-time control of parametric optimal control problems using boundary value methods. In Online Optimization of Large Scale Systems, Grötschel M et al. (eds). Springer: Berlin, 2001; 17-55.
4. Maurer H, Oberle HJ. Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. SIAM Journal on Control and Optimization 2002; 41:380-403.
5. Maurer H, Pickenhain S. Second order sufficient conditions for optimal control problems with mixed control-state constraints. Journal of Optimization Theory and Applications 1995; 86:649-667.
6. Milyutin AA, Osmolovskii NP. Calculus of variations and optimal control. Translations of Mathematical Monographs, vol. 180. American Mathematical Society: Providence, RI, 1998.
7. Zeidan V. The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency. SIAM Journal on Control and Optimization 1994; 32:1297-1321.
8. Noble J, Schättler H. Sufficient conditions for relative minima of broken extremals in optimal control theory. Journal of Mathematical Analysis and Applications 2002; 269:98-128.
9. ∞ nonsingular case. SIAM Journal on Control and Optimization 1987; 25:433-465.
10. Sussmann HJ. The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case. SIAM Journal on Control and Optimization 1987; 25:868-904.
11. Osmolovskii NP. High-order necessary and sufficient conditions for Pontryagin and bounded-strong minima in the optimal control problems. Doklady Akademii Nauk SSSR. Seriya - Cybernetics and Control Theory 1988; 303:1052-1056.
12. English transi., Soviet Physics Doklady 1988; 33(12):883-885.
13. Maurer H, Osmolovskii NP. Second order sufficient conditions for time-optimal bang-bang control problems. SIAM Journal on Control and Optimization 2004; 42:2239-2263.
14. Maurer H, Osmolovskii NP. Quadratic sufficient optimality conditions for bang-bang control problems. Control and Cybernetics 2003; 33:555-584.
15. Agrachev AA, Stefani G, Zezza PL. Strong optimality for a bang-bang trajectory. SIAM Journal on Control and Optimization 2002; 41(4):991-1014.
16. Osmolovskii NP, Maurer H. Equivalence of second order optimality conditions for bang-bang control problems. Part 1: main results. Control and Cybernetics, 2005, in press.
17. Kaya CY, Noakes JL. Computational method for time-optimal switching control. Journal of Optimization Theory and Applications 2003; 117:69-92.
18. Büskens C. Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen. Dissertation, Institut für Numerische Mathematik, Universität Münster, 1998.
19. Felgenhauer U. On stability of bang-bang type controls. SIAM Journal on Control and Optimization 2003; 41:1843-1867.
20. Kaya CY, Noakes JL. Computations of time-optimal controls. Optimal Control Applications and Methods 1996; 17:171-185.
21. Simakov ST, Kaya CY, Lucas SK. Computations for time-optimal bang-bang control using a Lagrangian formulation. Preprints of the 15th World Congress of IFAC, Barcelona, 2002.
22. Fiacco AV. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Mathematics in Science and Engineering, vol. 165. Academic Press: New York, 1983.
23. Büskens C, Maurer H. Sensitivity analysis and real-time optimization of parametric nonlinear programming problems. In Online Optimization of Large Scale Systems, Grötschel M et al. (eds). Springer: Berlin, 2001; 3-16.
24. Büskens C, Pesch HJ, Winderl S. Real-time solutions of bang-bang and singular optimal control problems. In Online Optimization of Large Scale Systems, Grötschel M et al. (eds). Springer: Berlin, 2001; 129-142.
25. Oberle HJ, Grimm W. BNDSCO - a program for the numerical solution of optimal control problems. Report 515-89122, Institute for Flight Systems Dynamics, DLR, Operpfaffenhofen, Germany, 1989.
26. Vassiliadis VS, Sargent RW, Pantelides CC. Solution of a class of multistage dynamic optimization problems. 2. Problems with path constraints. Industrial and Engineering Chemistry Research 1994; 33:2123-2133.
27. Chernousko FL, Akulenko LD, Bolotnik NN. Time-optimal control for robotic manipulators. Optimal Control Applications and Methods 1989; 10:293-311.
28. Geering HP, Guzella L, Hepner SAR, Onder CH. Time-optimal motions of robots in assembly tasks. IEEE Transactions on Automatic Control 1986; AC-31:512-518.
29. Göllmann L. Numerische Berechnung zeitoptimaler Trajektorien für zweigliedrige Roboterarme. Diploma Thesis, Institut für Numerische Mathematik, Universität Münster, 1991.
30. Oberle HJ. Numerical computation of singular control functions for a two-link robot arm. Lecture Notes in Control and Information Sciences 1987; 95:244-253.
31. Chyba M, Sussmann H, Maurer H, Vossen G. Underwater vehicles: the minimum time problem. Proceedings of 43rd IEEE Conference on Decision and Control, Bahamas, U.S.A., December 2004, to appear.
32. Büskens C, Maurer H. SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control. Journal of Computational and Applied Mathematics 2000; 120:85-108.
33. Kim JHR, Maurer H. Sensitivity analysis of optimal control problems with bang-bang controls. Proceedings of 42nd IEEE Conference on Decision and Control, Maui, U.S.A., December 2003; 3281-3286.
34. Kostyukova O, Kostina E. Analysis of properties of the solutions to parametric time-optimal problems. Computational Optimization and Applications 2003; 26:285-326.
35. Kim JHR. Optimierungsmethoden und Sensitivitätsanalyse f̈r optimale bang-bang Steuerungen mit Anwendungen in der Nichtlinearen Optik. Dissertation, Institut für Numerische Mathematik, Universität Münster, 2002.
36. Kim JHR, Lippi GL, Maurer H. Minimizing the transition time in lasers by optimal control methods. Single mode semiconductor laser with homogeneous transverse profile. Physica D - Nonlinear Phenomena 2004; 191:238-260.