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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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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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