Advertisement

Discretization of semilinear bang-singular-bang control problems

  • Ursula Felgenhauer
Article

Abstract

Bang-singular controls may appear in optimal control problems where the control enters the system linearly. We analyze a discretization of the first-order system of necessary optimality conditions written in terms of a variational inequality (or: inclusion) under appropriate assumptions including second-order optimality conditions. For the so-called semilinear case, it is proved that the discrete control has the same principal bang-singular-bang structure as the reference control and, in \(L_{1}\) topology, the convergence is of order one w.r.t. the stepsize.

Keywords

Bang-singular control structure Approximation of extremals Euler method \(L_{1}\) error estimate 

Mathematics Subject Classification

49M25 49M05 49J30 

Notes

Acknowledgments

The author is grateful to the anonymous referees for their instructive comments which, in particular, helped to close a gap in one of the main proofs.

References

  1. 1.
    Alt, W., Seydenschwanz, M.: Regularization and discretization of linear-quadratic control problems. Control Cybern. 40(4), 903–920 (2011)MathSciNetMATHGoogle Scholar
  2. 2.
    Alt, W., Seydenschwanz, M.: An implicit discretization scheme for linear-quadratic control problems with bang-bang solutions. Optim. Methods Softw. 29(3), 535–560 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alt, W., Baier, R., Gerdts, M., Lempio, F.: Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numer. Algebra Control Optim. 2(3), 547–570 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alt, W., Baier, R., Lempio, F., Gerdts, M.: Approximations of linear control problems with bang-bang solutions. Optimization 62(1), 9–32 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Aronna, S.M., 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Aronna, M.S., Bonnans, J.F., Martinon, P.: A shooting algorithm for optimal control problems with singular arcs. J. Optim. Theory Appl. 158(2), 419–459 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dmitruk, A.V.: Quadratic conditions for a weak minimium for singular regimes in optimal control problems. Sov. Math. Dokl. 18(2), 1 (1977)Google Scholar
  8. 8.
    Dmitruk, A.V.: Conditions of Jacobi type for Bolza’s problem with inequalities. Math. Notes 35, 427–435 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dmitruk, A.V.: Jacobi type conditions for singular extremals. Control Cybern. 37(2), 285–306 (2008)MathSciNetMATHGoogle Scholar
  10. 10.
    Dmitruk, A.V., Milyutin, A.A.: A condition of Legendre type for optimal control problems, linear in control. In: Loffe, A., Reich, S., Shafrir, I. (eds.) Calculus of Variations and Optimal Control (Haifa, 1998), pp. 49–61. Chapman & Hall/CRC, Boca Raton (2000)Google Scholar
  11. 11.
    Dobrowolski, M.: Angewandte Funktionalanalysis. Springer, Berlin (2006)Google Scholar
  12. 12.
    Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control. Math. Comput. 70(233), 173–203 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bounds for Euler approximation of a state and control constrained optimal control. Numer. Funct. Anal. Optim. 21, 653–682 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Felgenhauer, U.: Structural stability investigation of bang-singular-bang optimal controls. J. Optim. Theory Appl. 152(3), 605–631 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Felgenhauer, U.: Stability analysis of variational inequalities for bang-singular-bang controls. Control Cybern. 42(3), 557–592 (2013)MathSciNetMATHGoogle Scholar
  16. 16.
    Fischer, A.: An NCP-function and its use for the solution of complementarity problems. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 88–105. World Scientific Publishers, Minneapolis (1995)CrossRefGoogle Scholar
  17. 17.
    Fischer, A.: A Newton-type method for positive-semidefinite linear complementarity problems. J. Optim. Theory Appl. 46(3), 585–608 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fischer, A., Jiang, H.: Merit functions for complementarity and related problems: a survey. Comput. Optim. Appl. 17, 159–182 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gabasov, R., Kirillova, F.M.: High order necessary conditions for optimality. SIAM J. Control 10(1), 127–168 (1972)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Goh, B.S.: The second variation for the singular Bolza problem. SIAM J. Control 4(2), 309–325 (1966)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Goh, B.S.: Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4(4), 716–731 (1966)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Grigorieff, R.D.: Diskrete Approximation von Eigenwertproblemen, I. Qualitative Konvergenz. Numer. Math. 24(4), 355–374 (1975)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. (2) 20(4), 292–296 (1919)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Haunschmied, J.L., Pietrus, A., Veliov, V.M.: The Euler method for linear control systems revisited. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) Large-Scale Scientific Computing. Lecture Notes in Computer Science, pp. 90–97. Springer, Heidelberg (2014)Google Scholar
  25. 25.
    Jiang, H.: Smoothed Fischer-Burmeister equation methods for the complementarity problem, Technical Report, Department of Mathematics, The University of Melbourne, Melbourne (1997)Google Scholar
  26. 26.
    Kelley, H.J.: A second variation test for singular extremals. AIAA J. 2(8), 1380–1382 (1964)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)MATHGoogle Scholar
  28. 28.
    Korytowski, A., Szymkat, M., Maurer, H., Vossen, G.: Optimal control of a fedbatch fermentation process: numerical methods, sufficient conditions, and sensitivity analysis, In: 47th IEEE Conference on Decision and Control, Cancun (Mexico) 2008, IEEE, Control Systems Society, 1551–1556 (2008). doi:  10.1109/CDC.2008.4739078
  29. 29.
    Krener, A.J.: The high order maximum principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256–292 (1977)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ledzewicz, U., Schättler, H.: Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46(3), 1052–1079 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ledzewicz, U., Maurer, H., Schättler, H.: Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, In: 48th IEEE Conference on Decision and Control, Shanghai, China, IEEE Control Systems Society, 2280–2285 (2009). doi:  10.1109/CDC.2009.5400735
  32. 32.
    Malanowski, K., Büskens, C., Maurer, H.: Convergence of approximations to nonlinear control problems. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbation. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1997)Google Scholar
  33. 33.
    Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235–257 (1976)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Maurer, H., Tarnopolskaya, T., Fulton, N.L.: Singular controls in optimal collision avoidance for participants with unequal linear speeds. ANZIAM J. Electron. Suppl. 53(C), C1–C18 (2011)MathSciNetGoogle Scholar
  35. 35.
    Milyutin, A.A., Osmolovskii, N.P.: Calculus of Variations and Optimal Control. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  36. 36.
    Moyer, H.G.: Sufficient conditions for a strong minimum in singular control problems. SIAM J. Control 11(4), 620–636 (1973)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Poggiolini, L., Stefani, G.: Minimum time optimality for a bang-singular arc: second-order sufficient conditions, In: 44th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2005, Sevilla, 1433–1438 (2005). doi: 10.1109/CDC.2005.1582360
  38. 38.
    Poggiolini, L., Stefani, G.: Sufficient optimality conditions for a bang-singular extremal in the minimum time problem. Control Cybern. 37(2), 469–490 (2008)MathSciNetMATHGoogle Scholar
  39. 39.
    Poggiolini, L., Stefani, G.: Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17(4), 469–514 (2011)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Poggiolini, L., Stefani, G.: Structural stability for bang-singular-bang extremals in the minimum time problem. SIAM J. Control Optim. 51(5), 3511–3531 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Robinson, S.M.: Generalized equations and their solutions, I. Basic theory. In: Robinson, S.M. (ed.) Point-to-Set Maps and Mathematical Programming, Mathematical Programming Study, pp. 128–141. Dekker, New York (1979)CrossRefGoogle Scholar
  42. 42.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Robinson, S.M.: Generalized equations and their solutions, II. Applications to nonlinear programming. In: Robinson, S.M. (ed.) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, pp. 200–221. Dekker, New York (1982)CrossRefGoogle Scholar
  44. 44.
    Stefani, G.: Strong optimality of singular trajectories. In: Ancona, F., et al. (eds.) Geometric Control and Nonsmooth Analysis. Applied Mathematical Sciences Series, pp. 300–327. World Science Publisher, Hackensack (2008)CrossRefGoogle Scholar
  45. 45.
    Veliov, V.: Error analysis of discrete approximations to bang-bang optimal control problems: the linear case. Control Cybern. 34(3), 967–982 (2005)MathSciNetMATHGoogle Scholar
  46. 46.
    Veliov, V.: Approximations with error estimates for optimal control problems for linear system. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) Large-Scale Scientific Computing. Lecture Notes in Computer Science, pp. 263–270. Springer, Berlin (2006)CrossRefGoogle Scholar
  47. 47.
    Vossen, G.: Numerische Lösungsmethoden, hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für optimale bang-bang und singuläre Steuerungen, Ph.D. Thesis, Faculty of Mathematics and Science, University Münster (2005)Google Scholar
  48. 48.
    Vossen, G.: Switching time optimization for bang-bang and singular controls. J. Optim. Theory Appl. 144, 409–429 (2010)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Zelikin, M.I., Borisov, V.F.: Theory of Chattering Control. Birkhäuser, Boston (1994)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut für Angewandte Mathematik und Wissenschaftliches RechnenBrandenburgische Technische Universität Cottbus – SenftenbergCottbusGermany

Personalised recommendations