Optimal control for reconstruction of curves without cusps

Details Optimal control for reconstruction of curves without cusps Optimal control for reconstruction of curves without cusps

2012-03-14

arxiv.org/abs/1203.3089v1

Mathematics

Optimization and Control

Scientific paper

We consider the problem of minimizing $\int_0^L\sqrt{1+K(t)^2} dt$ for a planar curve having fixed initial and final positions and directions. Here $K(t)$ is the curvature of the curve and the total length $L$ is free. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that, depending on the boundary conditions, only two cases are possible: either there exists a global minimizer that is smooth and without cusps; or there is neither a global nor a local minimizer nor a geodesic. Our main tool is the construction of the optimal synthesis for the Reed and Shepp car with quadratic cost.

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