Second order optimality conditions in the smooth case and applications in optimal control
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 2, p. 207-236

The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity 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 the attitude control problem of a rigid spacecraft. This algorithm involves both normal and abnormal cases.

DOI : https://doi.org/10.1051/cocv:2007012
Classification:  49K15,  49-04,  70Q05
Keywords: conjugate point, second-order intrinsic derivative, lagrangian singularity, Jacobi field, orbit transfer, attitude control
@article{COCV_2007__13_2_207_0,
     author = {Bonnard, Bernard and Caillau, Jean-Baptiste and Tr\'elat, Emmanuel},
     title = {Second order optimality conditions in the smooth case and applications in optimal control},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {2},
     year = {2007},
     pages = {207-236},
     doi = {10.1051/cocv:2007012},
     zbl = {1123.49014},
     mrnumber = {2306634},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2007__13_2_207_0}
}
Bonnard, Bernard; Caillau, Jean-Baptiste; Trélat, Emmanuel. Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 2, pp. 207-236. doi : 10.1051/cocv:2007012. http://www.numdam.org/item/COCV_2007__13_2_207_0/

[1] A.A. Agrachev and R.V. Gamkrelidze, 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. | Zbl 0341.49007

[2] 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. | Zbl 0719.49023

[3] 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. | MR 2062547 | Zbl 1062.93001

[4] A.A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. Henri Poincaré 13 (1996) 635-690. | Numdam | Zbl 0866.58023

[5] A.A. Agrachev and A.V. Sarychev, On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estim. Cont. 8 (1998) 87-118. | Zbl 0826.49012

[6] C. Bischof, A. Carle, P. Kladem and A. Mauer, Adifor 2.0: Automatic Differentiation of Fortran 77 Programs. IEEE Comput. Sci. Engrg. 3 (1996) 18-32.

[7] O. Bolza, Calculus of variations. Chelsea Publishing Co., New York (1973).

[8] B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problem. SIAM J. Control Optim. 29 (1991) 1300-1321. | Zbl 0744.93033

[9] 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). | MR 2130101

[10] B. Bonnard and M. Chyba, The role of singular trajectories in control theory. Springer Verlag, New York (2003). | MR 1996448

[11] B. Bonnard and I. Kupka, 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. | Zbl 0779.49025

[12] B. Bonnard, J.-B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits. Discrete Contin. Dyn. Syst. 5 (2005) 929-956. | Zbl 1082.70014

[13] 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.

[14] J.B. Caillau, J. Noailles and J. Gergaud, 3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust. J. Opt. Theory Appl. 118 (2003) 541-565. | Zbl 1066.70016

[15] Y. Chitour, F. Jean and E. Trélat, Genericity results for singular trajectories. J. Diff. Geom. 73 (2006) 45-73. | Zbl 1102.53019

[16] J. De Morant, Contrôle en temps minimal des réacteurs chimiques discontinus. Ph.D. Thesis, Univ. Rouen (1992).

[17] S. Galot, D. Hulin and J. Lafontaine, Riemannian geometry. Springer-Verlag, Berlin (1987). | MR 909697 | Zbl 0636.53001

[18] B.S. Goh, Necessary conditions for singular extremals involving multiple control variables. SIAM J. Cont. 4 (1966) 716-731. | Zbl 0161.29004

[19] M.R. Hestenes, Application of the theory of quadratic forms in Hilbert spaces to the calculus of variations. Pac. J. Math. 1 (1951) 525-582. | Zbl 0045.20806

[20] M.R. Hestenes, Optimization theory - the finite dimensional case. Wiley (1975). | Zbl 0327.90015

[21] A.D. Ioffe and V.M. Tikhomirov, Theory of extremal problems. North-Holland Publishing Co., Amsterdam (1979). | MR 528295 | Zbl 0407.90051

[22] 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.

[23] A.J. Krener, The high-order maximum principle and its applications to singular extremals. SIAM J. Cont. Opt. 15 (1977) 256-293. | Zbl 0354.49008

[24] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The mathematical theory of optimal processes. Wiley Interscience (1962). | MR 166037 | Zbl 0117.31702

[25] A.V. Sarychev, 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. | Zbl 0484.49012

[26] 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). | Zbl 0349.65042

[27] E. Trélat, Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: COCV 6 (2001) 387-414. | Numdam | Zbl 0996.93009

[28] L.C. Young, Lectures on the calculus of variations and optimal control theory. Chelsea, New York (1980).

[29] O. Zarrouati, Trajectoires spatiales. CNES-Cepadues, Toulouse (1987).