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Second order optimality conditions in the smooth case and applications in optimal control

Published online by Cambridge University Press:  12 May 2007

Bernard Bonnard ,
Jean-Baptiste Caillau  and
Emmanuel Trélat
Bernard Bonnard
Affiliation:
Univ. Dijon, IMB, Bât. Mirande, 9 avenue Alain Savary, 21078 Dijon Cedex, France; [email protected]
Jean-Baptiste Caillau
Affiliation:
ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France; [email protected]
Emmanuel Trélat
Affiliation:
Univ. Orléans, UFR Sciences Mathématiques, Labo. MAPMO, UMR 6628, Route de Chartres, BP 6759, 45067 Orléans Cedex 2, France; [email protected]
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Abstract

The aim of this article is to present algorithms to compute the firstconjugate time along a smooth extremal curve, where the trajectoryceases to be optimal. It is based on recent theoretical developmentsof geometric optimal control, and the article contains a reviewof second order optimality conditions.The computations are related to a testof positivity of the intrinsic second order derivative or a test ofsingularity of the extremal flow. We derive an algorithm called COTCOT(Conditions of Order Two and COnjugate Times), available on the web,and apply it to the minimal time problem of orbit transfer, and to theattitude control problem of a rigid spacecraft.This algorithm involves both normal and abnormal cases.

Keywords

Conjugate pointsecond-order intrinsic derivativeLagrangian singularityJacobi fieldorbit transferattitude control
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 207 - 236
Copyright
© EDP Sciences, SMAI, 2007

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References

Agrachev, A.A. and Gamkrelidze, R.V., Second order optimality condition for the time optimal problem. Matem. Sbornik 100 (1976) 610–643. English transl. in: Math. USSR Sbornik 29 (1976) 547–576.
A.A. Agrachev and R.V. Gamkrelidze, Symplectic geometry for optimal control, Nonlinear controllability and optimal control. Dekker, New York, Monogr. Textbooks Pure Appl. Math. 133 (1990) 263–277.
A.A. Agrachev and Yu.L. Sachkov, Control theory from the geometric viewpoint, Encyclopedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin (2004) 412 pp.
Agrachev, A.A. and Sarychev, A.V., Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. Henri Poincaré 13 (1996) 635–690. CrossRef
Agrachev, A.A. and Sarychev, A.V., On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estim. Cont. 8 (1998) 87–118.
Bischof, C., Carle, A., Kladem, P. and Mauer, A., Adifor 2.0: Automatic Differentiation of Fortran 77 Programs. IEEE Comput. Sci. Engrg. 3 (1996) 18–32. CrossRef
O. Bolza, Calculus of variations. Chelsea Publishing Co., New York (1973).
Bonnard, B., Feedback equivalence for nonlinear systems and the time optimal control problem. SIAM J. Control Optim. 29 (1991) 1300–1321. CrossRef
B. Bonnard and J.-B. Caillau, Introduction to nonlinear optimal control, in Advances Topics in Control Systems Theory, Lecture Notes from FAP 2004, F. Lamnabhi-Lagarrigue, A. Loria, E. Panteley Eds., Springer, Berlin (2005).
B. Bonnard and M. Chyba, The role of singular trajectories in control theory. Springer Verlag, New York (2003).
Bonnard, B. and Kupka, I., Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. 5 (1993) 111–159. CrossRef
Bonnard, B., Caillau, J.-B. and Trélat, E., Geometric optimal control of elliptic Keplerian orbits. Discrete Contin. Dyn. Syst. 5 (2005) 929–956.
B. Bonnard, J.-B. Caillau and E. Trélat, Cotcot: short reference manual, ENSEEIHT-IRIT Technical Report RT/APO/05/1 (2005) www.n7.fr/apo/cotcot.
Caillau, J.B., Noailles, J. and Gergaud, J., 3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust. J. Opt. Theory Appl. 118 (2003) 541–565. CrossRef
Chitour, Y., Jean, F. and Trélat, E., Genericity results for singular trajectories. J. Diff. Geom. 73 (2006) 45–73.
J. de Morant, Contrôle en temps minimal des réacteurs chimiques discontinus. Ph.D. Thesis, Univ. Rouen (1992).
S. Galot, D. Hulin and J. Lafontaine, Riemannian geometry. Springer-Verlag, Berlin (1987).
Goh, B.S., Necessary conditions for singular extremals involving multiple control variables. SIAM J. Cont. 4 (1966) 716–731. CrossRef
Hestenes, M.R., Application of the theory of quadratic forms in Hilbert spaces to the calculus of variations. Pac. J. Math. 1 (1951) 525–582.
M.R. Hestenes, Optimization theory – the finite dimensional case. Wiley (1975).
A.D. Ioffe and V.M. Tikhomirov, Theory of extremal problems. North-Holland Publishing Co., Amsterdam (1979).
H.J. Kelley, R. Kopp and H.G. Moyer, Singular extremals, in Topics in optimization, G. Leitman Ed., Academic Press, New York (1967) 63–101.
Krener, A.J., The high-order maximum principle and its applications to singular extremals. SIAM J. Cont. Opt. 15 (1977) 256–293. CrossRef
L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The mathematical theory of optimal processes. Wiley Interscience (1962).
Sarychev, A.V., The index of second variation of a control system. Matem. Sbornik 113 (1980) 464–486. English transl. in: Math. USSR Sbornik 41 (1982) 383–401.
L.F. Shampine, H.A. Watts and S. Davenport, Solving non-stiff ordinary differential equations – the state of the art. Technical Report sand75-0182, Sandia Laboratories, Albuquerque, New Mexico (1975).
Trélat, E., Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: COCV 6 (2001) 387–414. CrossRef
L.C. Young, Lectures on the calculus of variations and optimal control theory. Chelsea, New York (1980).
O. Zarrouati, Trajectoires spatiales. CNES-Cepadues, Toulouse (1987).