Advertisement

Normality and Nondegeneracy for Optimal Control Problems with State Constraints

  • Fernando A. C. C. Fontes
  • Hélène Frankowska
Article

Abstract

In this paper, we investigate normal and nondegenerate forms of the maximum principle for optimal control problems with state constraints. We propose new constraint qualifications guaranteeing nondegeneracy and normality that have to be checked on smaller sets of points of an optimal trajectory than those in known sufficient conditions. In fact, the constraint qualifications proposed impose the existence of an inward pointing velocity just on the instants of time for which the optimal trajectory has an outward pointing velocity.

Keywords

Optimal control Maximum principle State constraints Constraint qualifications Normality Degeneracy Nonsmooth analysis Oriented distance 

Mathematics Subject Classification

49K15 

Notes

Acknowledgments

The partial supports of projects FP7-ITN-264735-SADCO “Sensitivity Analysis for Deterministic Controller Design” and FCT/FEDER Project PTDC/EEA-CRO/116014/2009 “Optimal Control in Constrained and Hybrid Nonlinear Systems”, PTDC/EEI-AUT/1450/2012 “Optimal Control: Health, Energy and Robotics Applications” are gratefully acknowledged.

References

  1. 1.
    Ferreira, M.M.A.: On the regularity of optimal controls for a class of problems with state constraints. Int. J. Syst. Sci. 37(8), 495–502 (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Frankowska, H., Marchini, E.M.: Lipschitzianity of optimal trajectories for the Bolza optimal control problem. Calc. Var. Partial. Differ. Equ. 27(4), 467–492 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Galbraith, G.N., Vinter, R.B.: Lipschitz continuity of optimal controls for state constrained problems. SIAM J. Control Optim. 42(5), 1727–1744 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Frankowska, H., Hoehener, D., Tonon, D.: A second-order maximum principle in optimal control under state constraints. Serdica Math. J. 39(3–4), 233–270 (2013)MathSciNetGoogle Scholar
  5. 5.
    Hoehener, D.: Variational approach to second-order optimality conditions for control problems with pure state constraints. SIAM J. Control Optim. 50, 1139–1173 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hoehener, D.: Feasible perturbations of control systems with pure state constraints and applications to second-order optimality conditions. Appl. Math. Optim. 68, 219–253 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer, Boston (2000)CrossRefGoogle Scholar
  8. 8.
    Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optim. 35, 930–952 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Arutyunov, A.V., Tynyanskiy, N.T.: The maximum principle in a problem with phase constraints. Sov. J. Comput. Syst. Sci. 23, 28–35 (1985)MathSciNetGoogle Scholar
  10. 10.
    Dubovitskii, A.Y., Dubovitskii, V.A.: Necessary conditions for strong minimum in optimal control problems with degeneration of endpoint and phase constraints. Usp. Mat. Nauk 40(2), 175–176 (1985)MathSciNetGoogle Scholar
  11. 11.
    Ferreira, M.M.A., Vinter, R.B.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl. 187(2), 438–467 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Ferreira, M.M.A., Fontes, F.A.C.C., Vinter, R.B.: Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints. J. Math. Anal. Appl. 233, 116–129 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lopes, S.O., Fontes, F.A.C.C., de Pinho, M.R.: On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete Contin. Dyn. Syst. A 29(2), 559–576 (2011)zbMATHCrossRefGoogle Scholar
  14. 14.
    Lopes, S.O., Fontes, F.A.C.C., de Pinho, M.R.: An integral-type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints. Syst. Control Lett. 62, 686–692 (2013)zbMATHCrossRefGoogle Scholar
  15. 15.
    Rampazzo, F., Vinter, R.B.: Degenerate optimal control problems with state constraints. SIAM J. Control Optim. 39, 989–1007 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Frankowska, H.: Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern. 38(4B), 1327–1340 (2009)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Frankowska, H., Mazzola, M.: On relations of the adjoint state to the value function for optimal control problems with state constraints. Nonlinear Differ. Equ. Appl. 20, 361–383 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Cernea, A., Frankowska, H.: A connection between the maximum principle and dynamic programming for constrained control problems. SIAM J. Control 44, 673–703 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Frankowska, H., Mazzola, M.: Optimal synthesis and normality of the maximum principle for optimal control problems with pure state constraints. In: Proceedings of 9th IEEE International Conference on Control and Automation (ICCA), Santiago, Chile, pp. 945–950 (2011)Google Scholar
  20. 20.
    Frankowska, H., Tonon, D.: Inward pointing trajectories, normality of the maximum principle and the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints. J. Convex Anal. 20(4), 1147–1180 (2013)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Lopes, S.O., Fontes, F.A.C.C.: Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints. J. Math. Anal. Appl. 399, 27–37 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)zbMATHGoogle Scholar
  23. 23.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T., Wets, R.B.: Variational Analysis. Grundlehren Math. Wiss., vol. 317. Springer, Berlin (1998)CrossRefGoogle Scholar
  25. 25.
    Clarke, F.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)zbMATHCrossRefGoogle Scholar
  26. 26.
    Frankowska, H., Mazzola, M.: Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints. Calc. Var. Partial. Differ. Equ. 46, 725–747 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems under constraints. Dokl. Akad. Nauk SSSR 149, 759–762 (1963)Google Scholar
  28. 28.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)zbMATHGoogle Scholar
  29. 29.
    Vinter, R.B., Pappas, G.: A maximum principle for nonsmooth optimal-control problems with state constraints. J. Math. Anal. Appl. 89(1), 212–232 (1982)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Fernando A. C. C. Fontes
    • 1
  • Hélène Frankowska
    • 2
  1. 1.SYSTEC-ISR, Faculdade de EngenhariaUniversidade do PortoPortoPortugal
  2. 2.CNRS, IMJ-PRG, UMR 7586Sorbonne Universits, UPMC Univ Paris 06, Univ Paris DiderotParisFrance

Personalised recommendations