A Shooting Algorithm for Optimal Control Problems with Singular Arcs (original) (raw)

References

  1. Goodman, T.R., Lance, G.N.: The numerical integration of two-point boundary value problems. Math. Tables Other Aids Comput. 10, 82–86 (1956)
    Article MathSciNet MATH Google Scholar
  2. Morrison, D.D., Riley, J.D., Zancanaro, J.F.: Multiple shooting method for two-point boundary value problems. Commun. ACM 5, 613–614 (1962)
    Article MathSciNet MATH Google Scholar
  3. Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Ginn-Blaisdell, Waltham (1968)
    MATH Google Scholar
  4. Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Technical report, Carl-Cranz-Gesellschaft, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Oberpfaffenhofen, Germany (1971)
  5. Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235–257 (1976)
    Article MATH Google Scholar
  6. Oberle, H.J.: Numerische Behandlung singulärer Steuerungen mit der Mehrzielmethode am Beispiel der Klimatisierung von Sonnenhäusern. Ph.D. thesis, Technische Universität München (1977)
  7. Oberle, H.J.: Numerical computation of singular control problems with application to optimal heating and cooling by solar energy. Appl. Math. Optim. 5(4), 297–314 (1979)
    Article MathSciNet MATH Google Scholar
  8. Fraser-Andrews, G.: Finding candidate singular optimal controls: a state-of-the-art survey. J. Optim. Theory Appl. 60(2), 173–190 (1989)
    Article MathSciNet MATH Google Scholar
  9. Martinon, P.: Numerical resolution of optimal control problems by a piecewise linear continuation method. Ph.D. thesis, Institut National Polytechnique de Toulouse (2005). Online: http://www.cmap.polytechnique.fr/~martinon/docs/Martinon-Thesis.pdf
  10. Vossen, G.: Switching time optimization for bang-bang and singular controls. J. Optim. Theory Appl. 144(2), 409–429 (2010)
    Article MathSciNet MATH Google Scholar
  11. Aronna, M.S.: Singular solutions in optimal control: second order conditions and a shooting algorithm. Research report Nr. 7764, INRIA (2011)
  12. Bonnard, B., Kupka, I.: Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. 5(2), 111–159 (1993)
    MathSciNet MATH Google Scholar
  13. Bonnard, B., Caillau, J.B., Trélat, E.: Second order optimality conditions in the smooth case and applications in optimal control. ESAIM Control Optim. Calc. Var. 13(2), 207–236 (2007) (electronic)
    Article MathSciNet MATH Google Scholar
  14. Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathematics & Applications, vol. 40. Springer, Berlin (2003)
    MATH Google Scholar
  15. Malanowski, K., Maurer, H.: Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl. 5, 253–283 (1996)
    Article MathSciNet MATH Google Scholar
  16. Bonnans, J.F., Hermant, A.: Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Annals of I.H.P. Nonlinear Anal. 26, 561–598 (2009)
    MathSciNet MATH Google Scholar
  17. Dennis, J.E.: Nonlinear least-squares. In: Jacobs, D. (ed.) The State-of-the-Art in Numerical Analysis, pp. 269–312. Academic Press, London (1977)
    Google Scholar
  18. Fletcher, R.: Practical methods of optimization. In: Unconstrained Optimization. A Wiley-Interscience Publication, vol. 1. Wiley, Chichester (1980)
    Google Scholar
  19. Dennis, J.E., Gay, D.M., Welsch, R.E.: An adaptive nonlinear least-squares algorithm. ACM Trans. Math. Softw. 7, 348–368 (1981)
    Article MATH Google Scholar
  20. Dmitruk, A.V.: Quadratic weak minimum conditions for special situations in optimal control problems. Dokl. Akad. Nauk SSSR 233(4), 523–526 (1977)
    MathSciNet Google Scholar
  21. Dmitruk, A.V.: Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control. Math. USSR, Izv. 28, 275–303 (1987)
    Article MATH Google Scholar
  22. Felgenhauer, U.: Structural stability investigation of bang-singular-bang optimal controls. J. Optim. Theory Appl. 152(3), 605–631 (2012)
    Article MathSciNet MATH Google Scholar
  23. Felgenhauer, U.: Controllability and stability for problems with bang-singular-bang optimal control (2011, submitted)
  24. Pontryagin, L., Boltyanski, V., Gamkrelidze, R., Michtchenko, E.: The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York (1962)
    MATH Google Scholar
  25. Kelley, H.J.: A second variation test for singular extremals. AIAA J. 2, 1380–1382 (1964)
    Article MathSciNet MATH Google Scholar
  26. Goh, B.S.: The second variation for the singular Bolza problem. SIAM J. Control 4(2), 309–325 (1966)
    Article MathSciNet MATH Google Scholar
  27. Goh, B.S.: Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4, 716–731 (1966)
    Article MathSciNet MATH Google Scholar
  28. Goh, B.S.: Necessary conditions for the singular extremals in the calculus of variations. Ph.D. thesis, University of Canterbury, New Zealand (1966)
  29. Kelley, H.J., Kopp, R.E., Moyer, H.G.: Singular extremals. In: Topics in Optimization, pp. 63–101. Academic Press, New York (1967)
    Chapter Google Scholar
  30. Robbins, H.M.: A generalized Legendre–Clebsch condition for the singular case of optimal control. IBM J. Res. Dev. 11, 361–372 (1967)
    Article MATH Google Scholar
  31. Bonnans, J.F.: Optimisation Continue. Dunod (2006)
    Google Scholar
  32. Levitin, E.S., Milyutin, A.A., Osmolovskiĭ, N.P.: Higher order conditions for local minima in problems with constraints. Usp. Mat. Nauk 33(6(204)), 85–148 (1978). 272
    MATH Google Scholar
  33. Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.A.: Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim. 2(3), 511–546 (2012). Special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday
    Article MathSciNet MATH Google Scholar
  34. Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, New York (1975)
    MATH Google Scholar
  35. Zeidan, V.: Sufficiency criteria via focal points and via coupled points. SIAM J. Control Optim. 30(1), 82–98 (1992)
    Article MathSciNet MATH Google Scholar
  36. Bonnard, B., Caillau, J.-B., Trélat, E.: Cotcot: Short reference manual. Technical report RT/APO/05/1, ENSEEIHT-IRIT, (2005)
  37. Fuller, A.T.: Study of an optimum non-linear control system. J. Electron. Control 15, 63–71 (1963)
    Article MathSciNet Google Scholar
  38. Betts, J.T.: Survey of numerical methods for trajectory optimization. AIAA J. Guid. Control Dyn. 21, 193–207 (1998)
    Article MATH Google Scholar
  39. Biegler, L.T.: Nonlinear programming: concepts, algorithms, and applications to chemical processes. In: MOS-SIAM Series on Optimization. SIAM, Philadelphia (2010)
    Google Scholar
  40. Gergaud, J., Martinon, P.: An application of PL continuation methods to singular arcs problems. In: Seeger, A. (ed.) Recent Advances in Optimization. Lectures Notes in Economics and Mathematical Systems, vol. 563, pp. 163–186. Springer, Berlin (2006)
    Chapter Google Scholar
  41. Bonnans, J.F., Martinon, P., Trélat, E.: Singular arcs in the generalized Goddard’s problem. J. Optim. Theory Appl. 139(2), 439–461 (2008)
    Article MathSciNet Google Scholar
  42. Clark, C.W.: Mathematical Bioeconomics. Wiley, New York (1976)
    MATH Google Scholar
  43. Aly, G.M.: The computation of optimal singular control. Int. J. Control 28(5), 681–688 (1978)
    Article MathSciNet MATH Google Scholar
  44. Goddard, R.H.: A method of reaching extreme altitudes. In: Smithsonian Miscellaneous Collections, vol. 71, No. 2. Smithsonian Institution, Washington (1919)
    Google Scholar
  45. Seywald, H., Cliff, E.M.: Goddard problem in presence of a dynamic pressure limit. J. Guid. Control Dyn. 16(4), 776–781 (1993)
    Article MATH Google Scholar
  46. Martinon, P., Gergaud, J.: Shoot2.0: An indirect grid shooting package for optimal control problems, with switching handling and embedded continuation. Research report Nr. 7380, INRIA, (2011)
  47. Garbow, B.S., Hillstrom, K.E., More, J.J.: User Guide for Minpack-1. National Argonne Laboratory, Illinois (1980)

Download references