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Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint

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Abstract

This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points.

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References

  1. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol. 45. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2003). Reprint of the 1990 edition [Springer, Berlin]

    MATH  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon/Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. Augustin, D., Maurer, H.: Computational sensitivity analysis for state constrained optimal control problems. Ann. Oper. Res. 101, 75–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berkmann, P., Pesch, H.J.: Abort landing in windshear: optimal control problem with third-order state constraint and varied switching structure. J. Optim. Theory Appl. 85 (1995)

  5. Bonnans, J.F., Hermant, A.: Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control. SIAM J. Control Optim. 46(4), 1398–1430 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bonnans, J.F., Hermant, A.: Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods. ESAIM: Control Optim. Calc. Var. 14(4), 825–863 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bonnans, J.F., Hermant, A.: No gap second order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. 117, 21–50 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonnans, J.F., Hermant, A.: Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. H. Poincaré (C) Anal. Nonlinéaire 26(2), 561–598 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bonnard, B., Faubourg, L., Trelat, E.: Optimal control of the atmospheric arc of a space shuttle and numerical simulations with multiple-shooting method. Math. Mod. Methods Appl. Sci. 15(1), 109–140 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bryson, A.E., Denham, W.F., Dreyfus, S.E.: Optimal programming problems with inequality constraints I: necessary conditions for extremal solutions. AIAA J. 1, 2544–2550 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bulirsch, R., Montrone, F., Pesch, H.J.: Abort landing in the presence of windshear as a minimax optimal control problem. II. Multiple shooting and homotopy. J. Optim. Theory Appl. 70(2), 223–254 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dontchev, A.L., Hager, W.W.: Lipschitzian stability for state constrained nonlinear optimal control. SIAM J. Control Optim. 36(2), 698–718 (1998) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control. Math. Comput. 70, 173–203 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Gergaud, J., Haberkorn, T.: Homotopy method for minimum consumption orbit transfer problem. ESAIM Control Optim. Calc. Var. 12(2), 294–310 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hager, W.W.: Lipschitz continuity for constrained processes. SIAM J. Control Optim. 17(3), 321–338 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hermant, A.: Optimal control of the atmospheric reentry of a space shuttle by an homotopy method. INRIA Research Report RR-6627 (2008). http://hal.inria.fr/inria-00317722/fr/

  17. Hermant, A.: Stability analysis of optimal control problems with a second-order state constraint. SIAM J. Optim. 20(1), 104–129 (2009)

    Article  MathSciNet  Google Scholar 

  18. Malanowski, K.: Stability and sensitivity of solutions to nonlinear optimal control problems. J. Appl. Math. Optim. 32, 111–141 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Malanowski, K.: Sufficient optimality conditions in stability analysis for state-constrained optimal control. Appl. Math. Optim. 55(2), 255–271 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Malanowski, K., Maurer, H.: Sensitivity analysis for state constrained optimal control problems. Discrete Contin. Dyn. Syst. 4, 241–272 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Malanowski, K., Maurer, H.: Sensitivity analysis for optimal control problems subject to higher order state constraints. Ann. Oper. Res. 101, 43–73 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Martinon, P., Gergaud, J.: An application of PL continuation methods to singular arcs problems. In: Recent Advances in Optimization. Lecture Notes in Econom. and Math. Systems, vol. 563, pp. 163–186. Springer, Berlin (2006)

    Chapter  Google Scholar 

  23. Maurer, H.: On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrum 41, Universität Münster (1979)

  24. Pesch, H.J.: A practical guide to the solution of real-life optimal control problems. Control Cybern. 23, 7–60 (1994)

    MATH  MathSciNet  Google Scholar 

  25. Robbins, H.M.: Junction phenomena for optimal control with state-variable inequality constraints of third order. J. Optim. Theory Appl. 31, 85–99 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1993)

    MATH  Google Scholar 

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  1. CMAP, Ecole Polytechnique, INRIA Saclay Île-de-France, Route de Saclay, 91128, Palaiseau, France

    Audrey Hermant

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  1. Audrey Hermant
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Correspondence to Audrey Hermant.

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Hermant, A. Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint. Appl Math Optim 61, 85 (2010). https://doi.org/10.1007/s00245-009-9076-y

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