Minimization of length and curvature on planar curves

Abstract— In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional R √ 1 + K2 ds, depending both on 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 non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the “timereparameterized” functional R ˙γ(t) p 1 + K2 γ dt for all boundary conditions if initial and final directions are considered regardless to orientation.

• Type: Conference
• Year: 2009
• Where: CDC
• Authors: Ugo V. Boscain, Gregoire Charlot, Francesco Rossi
• Keywords: CDC 2009, Control Systems, Curvature K, Various Functional Spaces, final Directions