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Existence of planar curves minimizing length and curvature

  • Ugo Boscain
  • Grégoire Charlot
  • Francesco Rossi
Article

Abstract

We consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional \( \smallint \sqrt {1 + K_\gamma ^2 ds} \) , depending both on the length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find nonexistence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories may converge to curves with angles. We instead prove the existence of minimizers for the “time-reparametrized” functional \( \smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt} \) for all boundary conditions if the initial and final directions are considered regardless of orientation. In this case, minimizers may present cusps (at most two) but not angles.

Keywords

Optimal Control Problem STEKLOV Institute Planar Curf Minimum Time Problem Lavrentiev Phenomenon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint (Springer, Berlin, 2004), Encycl. Math. Sci. 87.MATHGoogle Scholar
  2. 2.
    G. Bellettini, “Variational Approximation of Functionals with Curvatures and Related Properties,” J. Convex Anal. 4(1), 91–108 (1997).MATHMathSciNetGoogle Scholar
  3. 3.
    U. Boscain and F. Rossi, “Invariant Carnot-Caratheodory Metrics on S 3, SO(3), SL(2), and Lens Spaces,” SIAM J. Control Optim. 47(4), 1851–1878 (2008).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    U. Boscain and F. Rossi, “Projective Reeds-Shepp Car on S 2 with Quadratic Cost,” ESAIM: Control Optim. Calc. Var. 16(2), 275–297 (2010).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    F. Cao, Y. Gousseau, S. Masnou, and P. Pérez, “Geometrically Guided Exemplar-Based Inpainting,” Preprint (2010), http://math.univ-lyon1.fr/homes-www/masnou/fichiers/publications/inpainting9.pdf
  6. 6.
    G. Citti and A. Sarti, “A Cortical Based Model of Perceptual Completion in the Roto-translation Space,” J. Math. Imaging Vision 24(3), 307–326 (2006).CrossRefMathSciNetGoogle Scholar
  7. 7.
    I. D. Coope, “Curve Interpolation with Nonlinear Spiral Splines,” IMA J. Numer. Anal. 13(3), 327–341 (1993).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    L. E. Dubins, “On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents,” Am. J. Math. 79(3), 497–516 (1957).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Linnér, “Existence of Free Nonclosed Euler-Bernoulli Elastica,” Nonlinear Anal., Theory Methods Appl. 21(8), 575–593 (1993).MATHCrossRefGoogle Scholar
  10. 10.
    A. Linnér, “Curve-Straightening and the Palais-Smale Condition,” Trans. Am. Math. Soc. 350(9), 3743–3765 (1998).MATHCrossRefGoogle Scholar
  11. 11.
    P. D. Loewen, “On the Lavrentiev Phenomenon,” Can. Math. Bull. 30(1), 102–108 (1987).MATHMathSciNetGoogle Scholar
  12. 12.
    I. Moiseev and Yu. L. Sachkov, “Maxwell Strata in Sub-Riemannian Problem on the Group of Motions of a Plane,” ESAIM: Control Optim. Calc. Var. 16(2), 380–399 (2010).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J. Petitot, Neurogéométrie de la vision: Modèles mathématiques et physiques des architectures fonctionnelles (Éd. École Polytech., Palaiseau, 2008).Google Scholar
  14. 14.
    J. Petitot and Y. Tondut, “Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux,” Math. Inform. Sci. Hum. 145, 5–101 (1999).MathSciNetGoogle Scholar
  15. 15.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Interscience, New York, 1962).MATHGoogle Scholar
  16. 16.
    J. A. Reeds and L. A. Shepp, “Optimal Paths for a Car That Goes both Forwards and Backwards,” Pac. J. Math. 145(2), 367–393 (1990).MathSciNetGoogle Scholar
  17. 17.
    Yu. L. Sachkov, “Conjugate and Cut Time in the Sub-Riemannian Problem on the Group of Motions of a Plane,” ESAIM: Control Optim. Calc. Var., 10.1051/cocv/2009031 (2009).Google Scholar
  18. 18.
    Yu. L. Sachkov, “Cut Time and Optimal Synthesis in Sub-Riemannian Problem on the Group of Motions of a Plane,” arXiv: 0903.0727.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • Ugo Boscain
    • 1
    • 2
  • Grégoire Charlot
    • 3
  • Francesco Rossi
    • 2
  1. 1.CNRS CMAPÉcole PolytechniquePalaiseau CedexFrance
  2. 2.SISSATriesteItaly
  3. 3.UMR5582Institut FourierSt. Martin d’Hères cedexFrance

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