Switching time optimization for bang-bang and singular controls

Journal of Optimization Theory and Applications

Volumn 144, Issue 2, 2010, Pages 409-429

  Vossen, G ^a  

^a RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Bang bang control; Second order sufficient conditions; Singular control; Variational derivatives

Indexed keywords

BANG BANG CONTROL SYSTEMS; OPTIMAL CONTROL SYSTEMS;

BANG-BANG CONTROL; CONTROL VARIABLE; FIRST ORDER; OPTIMAL CONTROL PROBLEM; SECOND-ORDER SUFFICIENT CONDITIONS; SINGULAR CONTROL; STATE TRAJECTORY; SWITCHING TIME; TIME OPTIMIZATION; VARIATIONAL DERIVATIVE;

NUMERICAL METHODS;

EID: 76149099867     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-009-9594-4     Document Type: Article

References (35)

1. Maurer, H.: First and second-order sufficient optimality conditions in mathematical programming and optimal control. Math. Program. Stud. 10, 163-177 (1981).
2. Pickenhain, S.: Sufficiency conditions for weak local minima in multidimensional optimal control problems with mixed control-state restrictions. Z. Anal. Anwend. 11, 559-568 (1992).
3. Zeidan, V.: The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency. SIAM J. Control Optim. 32, 1297-1321 (1994).
4. Boltyanskii, V. G.: Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM J. Control Optim. 4, 326-361 (1966).
5. Piccoli, B., Sussmann, H. J.: Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim. 39, 359-410 (2001).
6. 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).
7. Agrachev, A. A., Stefani, G., Zezza, P. L.: Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 41(4), 991-1014 (2002).
8. Osmolovskii, N. P., Maurer, H.: Equivalence of second-order optimality conditions for bang-bang control problems, part 1. Control Cybern. 34(3), 927-950 (2005).
9. Osmolovskii, N. P., Maurer, H.: Equivalence of second-order optimality conditions for bang-bang control problems, part 2. Control Cybern. 36(1), 5-46 (2007).
10. Dmitruk, A. V.: Quadratic order conditions of a local minimum for singular extremals in a general optimal control problem. Proc. Symp. Pure Math. 64, 163-198 (1999).
11. Poggiolini, L., Stefani, G.: Minimum time optimality for a bang-singular arc: second-order sufficient conditions. In: Proceedings of the 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference, CDC-ECC '05, Seville, Spain, pp. 1433-1438 (2005).
12. Vossen, G.: Numerische Lösungsmethoden, hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für optimale bang-bang und singuläre Steuerungen. PhD Thesis, Universität Münster, Germany (2006).
13. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F.: The Mathematical Theory of Optimal Processes. Fizmatgiz, Moscow (1961). English translation: Pergamon Press, New York (1964).
14. Vossen, G.: Switching time optimization for bang-bang and singular controls: variational derivatives and applications. Technical Report, Lehr- und Forschungsanstalt für Nichtlineare Dynamik, RWTH Aachen. Online: http://www. nld. rwth-aachen. de/de/139474. html.
15. Bryson, A. E., Ho, Y. C.: Applied Optimal Control. Wiley, New York (1975).
16. Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235-257 (1976).
17. Wächter, A., Biegler, L. T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25-57 (2006).
18. Büskens, C.: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen. PhD Thesis, Universität Münster, Germany (1998).
19. Milyutin, A. A., Osmolovskii, N. P.: Calculus of Variations and Optimal Control. Transl. Math. Monogr., vol. 180. AMS, Providence (1998).
20. Szymkat, M., Korytowski, A.: Method of monotone structural evolution for control and state constrained optimal control problems. In: Proceedings of the European Control Conference ECC 2003, University of Cambridge, U. K., September 1-4, 2003.
21. Kelley, H. J., Kopp, R. E., Moyer, H. G.: Singular extremals. In: Leitmann, G. (ed.) Topics in Optimization, pp. 63-101. Academic Press, New York (1967).
22. Bell, D. J., Jacobson, D. H.: Singular Optimal Control Problems. Academic Press, London (1975).
23. Fraser-Andrews, G.: Finding candidate singular optimal controls: a state of the art survey. J. Optim. Theory Appl. 60(2), 173-190 (1989).
24. McDanell, J. P., Powers, W. F.: Necessary conditions for joining singular and nonsingular subarcs. SIAM J. Control 9(2), 161-173 (1971).
25. Krener, A. J.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256-293 (1977).
26. Vossen, G., Maurer, H.: On L1-minimization in optimal control and applications to robotics. Optim. Control Appl. Methods 27, 301-321 (2006).
27. Chyba, M., Sussmann, H., Maurer, H., Vossen, G.: Underwater vehicles: the minimum time problem. In: Proceedings of the 43rd IEEE Conference on Decision and Control, CDC'04, The Bahamas, pp. 1370-1375. IEEE Control Soc. (2004).
28. Fiacco, A. V., McCormick, G. P.: Nonlinear Programming. Wiley, New York (1968).
29. Kaya, C. Y., Noakes, J. L.: Computational method for time-optimal switching control. J. Optim. Theory Appl. 117, 69-92 (2003).
30. Maurer, H., Büskens, C., Kim, J.-H. R., Kaya, C. Y.: Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optim. Control Appl. Methods 26, 129-156 (2005).
31. Lee, H. W. J., Teo, K. L., Jennings, L. S., Rehbock, V.: Control parametrization enhancing technique for time optimal control problems. Dyn. Syst. Appl. 6(2), 243-261 (1997).
32. Vossen, G., Rehbock, V., Siburian, A.: Numerical solution methods for singular control with multiple state dependent forms. Optim. Methods Softw. 22(4), 551-559 (2007).
33. Büskens, C., Pesch, H. J., Winderl, S.: Real-time solutions to bang-bang and singular optimal control problems. In: Grötschel, M., Krumke, S. O., Rambau, J. (eds.) Online Optimization of Large Scale Problems, pp. 129-142. Springer, Berlin (2001).
34. James, E. M.: Time optimal control and the van der Pol oscillator. J. Appl. Math. 13(1), 67-81 (1974).
35. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1993).