Projective Reeds-Shepp car on S2 with quadratic cost
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 16, Issue: 2, page 275-297
- ISSN: 1292-8119
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topBoscain, Ugo, and Rossi, Francesco. "Projective Reeds-Shepp car on S2 with quadratic cost." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 275-297. <http://eudml.org/doc/250766>.
@article{Boscain2010,
abstract = {
Fix two points $x,\bar\{x\}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J[\gamma]=\int_0^T \left(\{\g\}_\{\gamma(t)\}(\dot\gamma(t),\dot\gamma(t))+
K^2_\{\gamma(t)\}\{\g\}_\{\gamma(t)\}(\dot\gamma(t),\dot\gamma(t)) \right) ~\{\rm d\}t$
along all smooth curves starting from x with direction η and ending in $\bar\{x\}$ with direction $\bar\eta$. 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.
},
author = {Boscain, Ugo, Rossi, Francesco},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carnot-Caratheodory distance; geometry of vision; lens spaces; global cut locus; Carnot-Carathéodory distance},
language = {eng},
month = {4},
number = {2},
pages = {275-297},
publisher = {EDP Sciences},
title = {Projective Reeds-Shepp car on S2 with quadratic cost},
url = {http://eudml.org/doc/250766},
volume = {16},
year = {2010},
}
TY - JOUR
AU - Boscain, Ugo
AU - Rossi, Francesco
TI - Projective Reeds-Shepp car on S2 with quadratic cost
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 275
EP - 297
AB -
Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J[\gamma]=\int_0^T \left({\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+
K^2_{\gamma(t)}{\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) \right) ~{\rm d}t$
along all smooth curves starting from x with direction η and ending in $\bar{x}$ with direction $\bar\eta$. 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.
LA - eng
KW - Carnot-Caratheodory distance; geometry of vision; lens spaces; global cut locus; Carnot-Carathéodory distance
UR - http://eudml.org/doc/250766
ER -
References
top- A. Agrachev, Methods of control theory in nonholonomic geometry, in Proc. ICM-94, Birkhauser, Zürich (1995) 1473–1483.
- A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Contr. Syst.2 (1996) 321–358.
- A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Sem. Mat. Univ. Politec. Torino56 (2001) 1–12.
- A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences87. Springer (2004).
- A. Bellaiche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, Progress in Mathematics144, Birkhäuser, Basel (1996) 1–78.
- B. Bonnard and M. Chyba, Singular trajectories and their role in control theory. Springer-Verlag, Berlin (2003).
- U. Boscain and B. Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds, SMAI43. Springer (2004).
- 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.
- 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.
- A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, Appl. Math. Series2. American Institute of Mathematical Sciences (2007).
- R.W. Brockett, Explicitly solvable control problems with nonholonomic constraints, in Proceedings of the 38th IEEE Conference on Decision and Control1 (1999) 13–16.
- Y. Chitour and M. Sigalotti, Dubins' problem on surfaces. I. Nonnegative curvature J. Geom. Anal.15 (2005) 565–587.
- Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves. J. Differential Geometry73 (2006) 45–73.
- 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.
- M. Gromov, Carnot-Caratheodory spaces seen from within, in Sub-Riemannian Geometry, Progress in Mathematics144, Birkhäuser, Basel (1996) 79–323.
- V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997).
- V. Jurdjevic, Optimal Control, Geometry and Mechanics, in Mathematical Control Theory, J. Bailleu and J.C. Willems Eds., Springer, New York (1999) 227–267.
- V. Jurdjevic, Hamiltonian Point of View on non-Euclidean Geometry and Elliptic Functions. System Control Lett.43 (2001) 25–41.
- J. Petitot, Vers une Neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, in Mathématiques, Informatique et Sciences Humaines145, Special issue, EHESS, Paris (1999) 5–101.
- L.S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko, The Mathematical Theory of Optimal Processes. John Wiley and Sons, Inc. (1961).
- 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.
- D. Rolfsen, Knots and links. Publish or Perish, Houston (1990).
- Yu.L. Sachkov, Maxwell strata in Euler's elastic problem. J. Dyn. Contr. Syst.14 (2008) 169–234.
- M. Spivak, A comprehensive introduction to differential geometry. Second edition, Publish or Perish, Inc., Wilmington, Del. (1979).
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