Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 327-346.
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We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.

DOI : 10.1016/j.anihpc.2018.05.005
Mots-clés : Geometric optimal control, Chattering, Fuller, Genericity 
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     title = {Time-optimal trajectories of generic control-affine systems have at worst iterated {Fuller} singularities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {327--346},
     publisher = {Elsevier},
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Boarotto, Francesco; Sigalotti, Mario. Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 327-346. doi : 10.1016/j.anihpc.2018.05.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2018.05.005/

[1] Abraham, R.; Robbin, J. Transversal Mappings and Flows. An Appendix by Al Kelley, W. A. Benjamin, Inc., New York–Amsterdam, 1967 | MR | Zbl

[2] Agrachev, A.A. On regularity properties of extremal controls, J. Dyn. Control Syst., Volume 1 (1995) no. 3, pp. 319–324 | DOI | MR | Zbl

[3] Agrachev, A.A. Some open problems, Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., vol. 5, Springer, Cham, 2014, pp. 1–13 | DOI | MR | Zbl

[4] Agrachev, A.A.; Gamkrelidze, R.V. Symplectic geometry for optimal control, Nonlinear Controllability and Optimal Control, vol. 133, 1990, pp. 263–277 | MR | Zbl

[5] Agrachev, A.A.; Gamkrelidze, R.V. Symplectic geometry for optimal control, Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., vol. 133, Dekker, New York, 1990, pp. 263–277 | MR | Zbl

[6] Agrachev, A.A.; Sachkov, Y.L. Control theory from the geometric viewpoint, Control Theory and Optimization, II, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004 | MR | Zbl

[7] Agrachev, A.A.; Sigalotti, M. On the local structure of optimal trajectories in R3 , SIAM J. Control Optim., Volume 42 (2003) no. 2, pp. 513–531 | DOI | MR | Zbl

[8] Bonnard, B.; Kupka, I. Generic properties of singular trajectories, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 14 (1997) no. 2, pp. 167–186 | DOI | Numdam | MR | Zbl

[9] Boscain, U.; Piccoli, B. Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, Mathematics & Applications, vol. 43, Springer-Verlag, Berlin, 2004 | MR | Zbl

[10] Bressan, A. The generic local time-optimal stabilizing controls in dimension 3, SIAM J. Control Optim., Volume 24 (1986) no. 2, pp. 177–190 | DOI | MR | Zbl

[11] Caponigro, M.; Ghezzi, R.; Piccoli, B.; Trélat, E. Regularization of chattering phenomena via bounded variation control, IEEE Trans. Autom. Control (2018) (in press) | DOI | MR | Zbl

[12] Chitour, Y.; Jean, F.; Trélat, E. Singular trajectories of control-affine systems, SIAM J. Control Optim., Volume 47 (2008) no. 2, pp. 1078–1095 | DOI | MR | Zbl

[13] Fuller, A.T. Study of an optimum nonlinear system, J. Electron. Control, Volume 15 (1963), pp. 63–71 | DOI | MR

[14] Goresky, M.; MacPherson, R. Stratified Morse Theory, vol. 14, Springer Science & Business Media, 2012

[15] Krener, A.J.; Schaettler, H. The structure of small time reachable sets in low dimension, SIAM J. Control Optim., Volume 27 (1989), pp. 120–147 | DOI | MR | Zbl

[16] Kupka, I. The ubiquity of Fuller's phenomenon, Nonlinear controllability and optimal control, Volume 133 (1990), pp. 313–350 | MR | Zbl

[17] Lobry, C. Contrôlabilité des systèmes non linéaires, SIAM J. Control Optim., Volume 8 (1970) no. 4, pp. 573 | MR | Zbl

[18] Marchal, C. Chattering arcs and chattering controls, J. Optim. Theory Appl., Volume 11 (1973), pp. 441–468 | DOI | MR | Zbl

[19] Piccoli, B. Regular time-optimal syntheses for smooth planar systems, Rend. Semin. Mat. Univ. Padova, Volume 95 (1996), pp. 59–79 | Numdam | MR | Zbl

[20] Piccoli, B.; Sussmann, H.J. Regular synthesis and sufficiency conditions for optimality, SIAM J. Control Optim., Volume 39 (2000) no. 2, pp. 359–410 | DOI | MR | Zbl

[21] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York–London, 1962 (translated from the Russian by K.N. Trirogoff, edited by L.W. Neustadt) | MR | Zbl

[22] Schättler, H. The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Control Optim., Volume 26 (1988) no. 4, pp. 899–918 | DOI | MR | Zbl

[23] Schättler, H. On the local structure of time-optimal bang–bang trajectories in R3 , SIAM J. Control Optim., Volume 26 (1988) no. 1, pp. 186–204 | DOI | MR | Zbl

[24] Sigalotti, M. Regularity properties of optimal trajectories of control systems with one input in dimension three, Sovrem. Mat. Prilozh., Volume 9 (2003), pp. 68–80 | MR | Zbl

[25] Sigalotti, M. Local regularity of optimal trajectories for control problems with general boundary conditions, J. Dyn. Control Syst., Volume 11 (2005), pp. 91–123 | DOI | MR | Zbl

[26] Sussmann, H. Time-optimal control in the plane, Feedback Control of Linear and Nonlinear Systems, 1982, pp. 244–260 | DOI | MR | Zbl

[27] Sussmann, H. The structure of time-optimal trajectories for single-input systems in the plane: the C∞ nonsingular case, SIAM J. Control Optim., Volume 25 (1987) no. 2, pp. 433–465 | DOI | MR

[28] Sussmann, H.; Sussmann, H.J. Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets, Nonlinear Controllability and Optimal Control, Marcel Dekker, New York, 1990, pp. 1–20 | MR | Zbl

[29] Sussmann, H.J. A weak regularity theorem for real analytic optimal control problems, Rev. Mat. Iberoam., Volume 2 (1986) no. 3, pp. 307–317 | DOI | MR | Zbl

[30] Sussmann, H.J. Envelopes, conjugate points, and optimal bang–bang extremals, Algebraic and Geometric Methods in Nonlinear Control Theory, Math. Appl., vol. 29, Reidel, Dordrecht, 1986, pp. 325–346 | MR | Zbl

[31] Zelikin, M.I.; Borisov, V.F. Theory of Chattering Control: with Applications to Astronautics, Robotics, Economics, and Engineering, Springer Science & Business Media, 2012

[32] Zelikin, M.I.; Lokutsievskiy, L.V.; Hildebrand, R. Geometry of neighborhoods of singular trajectories in problems with multidimensional control, Proc. Steklov Inst. Math., Volume 277 (2012) no. 1, pp. 67–83 | DOI | MR | Zbl

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