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Une approche géométrique du contrôle optimal de l'arcatmosphérique de la navette spatiale

Published online by Cambridge University Press:  15 September 2002

Bernard Bonnard  and
Emmanuel Trélat
Bernard Bonnard
Affiliation:
Université de Bourgogne, LAAO, Dijon, France ; bbonnard@u-bourgogne.fr.
Emmanuel Trélat
Affiliation:
Université Paris Sud, Orsay, France ; trelat@topolog.u-bourgogne.fr.
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Abstract

The aim of this article is to make some geometric remarks and some preliminary calculations in order to construct the optimal atmospheric arc of a spatial shuttle (problem of reentry on Earth or Mars Sample Return project). The system describing the trajectories is in dimension 6, the control is the bank angle and the cost is the total thermal flux. Moreover there are state constraints (thermal flux, normal acceleration and dynamic pressure). Our study is mainly geometric and is founded on the evaluation of the accessibility set taking into account the state constraints. We make an analysis of the extremals of the Minimum Principle in the non-constrained case, and give a version of the Minimum Principle adapted to deal with the state constraints.

Keywords

Contrôle optimal avec contraintes sur l'étatprincipes du minimummécanique célestearc atmosphérique.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 179 - 222
Copyright
© EDP Sciences, SMAI, 2002

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References

Baumann, H. et Oberle, H.J., Numerical computation of optimal trajectories for coplanar aeroassisted orbital transfer. J. Optim. Theory Appl. 107 (2000) 457-479. CrossRef
O. Bolza, Calculus of variations. Chelsea (1973).
F. Bonnans et G. Launay, Large scale direct optimal control applied to the re-entry problem. J. Guidance, Control and Dynamics 21 (1998) 996-1000. CrossRef
Bonnard, B. et Launay, G., Time minimal control of batch reactors. ESAIM: COCV 3 (1998) 407-467. CrossRef
Bonnard, B. et Kupka, I., Théorie des singularités de l'application entrée/sortie et optimalité des singulières. Forum Math. 5 (1993) 111-159. CrossRef
A. Bryson et Y.C. Ho, Applied optimal control. Hemisphere Pub. Corporation (1975).
Caillau, J.B. et Noailles, J., Coplanar control of a satellite around the Earth. ESAIM: COCV 6 (2001) 239-258. CrossRef
CNES, Mécanique spatiale. Cepadues Eds. (1993).
J.M. Coron et L. Praly, Guidage en rentrée atmosphérique, Rapport 415. CNES (2000).
Ekeland, I., Discontinuité des champs de vecteurs extrémaux en calcul des variations. Publ. Math. IHES 47 (1977) 5-32. CrossRef
A.D. Ioffe et V.M. Tikhomirov, Theory of extremal problems. North Holland (1979).
Jacobson, P.H. et al., New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. 35 (1971) 255-284. CrossRef
Krener, A.J. et Schättler, H., The structure of small time reachable sets in small dimensions. SIAM J. Control Optim. 27 (1989) 120-147. CrossRef
Kupka, I., Geometric theory of extremals in optimal control problems. Trans. Amer. Math. Soc. 299 (1987) 225-243.
Maurer, H., On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim. 15 (1977) 345-362. CrossRef
Miele, A., Recent advances in the optimization and guidance of aeroassisted orbital transfers. Acta Astronautica 38 (1996) 747-768. CrossRef
H.J. Pesch, A practical guide to the solution of real-life optimal control problems. Control Cybernet. 23 (1994).
V. Pontryagin et al., Méthodes mathématiques des processus optimaux. Eds. Mir (1974).
Schättler, H., The local structure of time-optimal trajectories in dimension 3 under generic conditions. SIAM J. Control Optim. 26 (1988) 899-918. CrossRef
Sussmann, H.J., The structure of time-optimal trajectories for single-input systems in the plane: The $C^\infty$ non singular case. SIAM J. Control Optim. 25 (1987) 856-905.