Projective Reeds-Shepp car on S2 with quadratic cost

Boscain, Ugo; Rossi, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 275-297.

Fix two points $x,\overline{x}\in S^2$ and two directions (without orientation) $\eta ,\overline{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J\left[\gamma \right]={\int }_0^T\left({}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))+K_{\gamma \left(t\right)}^2{}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))\right)\mathrm{d}t$ along all smooth curves starting from x with direction η and ending in $\overline{x}$ with direction $\overline{\eta }$. Here g is the standard riemannian metric on S² and $K_{\gamma }$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

[1] A. Agrachev, Methods of control theory in nonholonomic geometry, in Proc. ICM-94, Birkhauser, Zürich (1995) 1473-1483. | Zbl

[2] A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Contr. Syst. 2 (1996) 321-358. | Zbl

[3] A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Sem. Mat. Univ. Politec. Torino 56 (2001) 1-12. | Zbl

[4] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences 87. Springer (2004). | Zbl

[5] A. Bellaiche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, Progress in Mathematics 144, Birkhäuser, Basel (1996) 1-78. | Zbl

[6] B. Bonnard and M. Chyba, Singular trajectories and their role in control theory. Springer-Verlag, Berlin (2003). | Zbl

[7] U. Boscain and B. Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds, SMAI 43. Springer (2004). | Zbl

[8] U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and lens spaces. SIAM J. Contr. Opt. 47 (2008) 1851-1878. | Zbl

[9] U. Boscain, T. Chambrion and J.P. Gauthier, On the K+P problem for a three-level quantum system: Optimality implies resonance. J. Dyn. Contr. Syst. 8 (2002) 547-572. | Zbl

[10] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, Appl. Math. Series 2. American Institute of Mathematical Sciences (2007). | Zbl

[11] R.W. Brockett, Explicitly solvable control problems with nonholonomic constraints, in Proceedings of the 38th IEEE Conference on Decision and Control 1 (1999) 13-16.

[12] Y. Chitour and M. Sigalotti, Dubins' problem on surfaces. I. Nonnegative curvature J. Geom. Anal. 15 (2005) 565-587. | Zbl

[13] Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves. J. Differential Geometry 73 (2006) 45-73. | Zbl

[14] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24 (2006) 307-326. | Zbl

[15] M. Gromov, Carnot-Caratheodory spaces seen from within, in Sub-Riemannian Geometry, Progress in Mathematics 144, Birkhäuser, Basel (1996) 79-323. | Zbl

[16] V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997). | Zbl

[17] V. Jurdjevic, Optimal Control, Geometry and Mechanics, in Mathematical Control Theory, J. Bailleu and J.C. Willems Eds., Springer, New York (1999) 227-267. | Zbl

[18] V. Jurdjevic, Hamiltonian Point of View on non-Euclidean Geometry and Elliptic Functions. System Control Lett. 43 (2001) 25-41. | Zbl

[19] J. Petitot, Vers une Neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, in Mathématiques, Informatique et Sciences Humaines 145, Special issue, EHESS, Paris (1999) 5-101.

[20] L.S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko, The Mathematical Theory of Optimal Processes. John Wiley and Sons, Inc. (1961). | Zbl

[21] J.A. Reeds and L.A. Shepp, Optimal paths for a car that goes both forwards and backwards. Pacific J. Math. 145 (1990) 367-393.

[22] D. Rolfsen, Knots and links. Publish or Perish, Houston (1990). | Zbl

[23] Yu.L. Sachkov, Maxwell strata in Euler's elastic problem. J. Dyn. Contr. Syst. 14 (2008) 169-234. | Zbl

[24] M. Spivak, A comprehensive introduction to differential geometry. Second edition, Publish or Perish, Inc., Wilmington, Del. (1979). | Zbl