NDLI: Existence, Multiplicity, and Regularity for sub-Riemannian Geodesics by Variational Methods

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Giamb, Roberto Piccione, Paolo Giannoni, Fabio
Copyright Year	2002
Abstract	We develop a variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold. Given a Riemannian manifold (M,g), a smooth distribution $\Delta\subset T{M}$ of codimension one in M, a point $p\in{M}$, and a smooth immersion $\gamma:\R\rightarrow{M}$ with closed image in M and which is everywhere transversal to $\Delta$, we look for curves in M that are stationary with respect to the Riemannian energy functional among all of the absolutely continuous curves horizontal with respect to $\Delta$ and that join p and $\gamma$. If (M,g) is complete, such extremizers exist, and they are curves of class C2 characterized as the solutions of an integro-differential equation or by a system of ordinary differential equations. We present some results concerning a sort of exponential map relative to the integro-differential equation and some applications. In particular, we obtain that if p and $gamma$ are sufficiently close in M, then there exists a unique length minimizer. We obtain existence and multiplicity results by means of the Ljusternik--Schnirelman theory.
Starting Page	1840
Ending Page	1857
Page Count	18
File Format	PDF
ISSN	03630129
DOI	10.1137/S0363012900367242
e-ISSN	10957138
Issue Number	6
Volume Number	40
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-26
Access Restriction	Subscribed
Subject Keyword	Applications to control theory Applications to the theory of geodesics Sub-Riemannian geometry Geodesics Abstract critical point theory sub-Riemannian geodesics Ljusternik--Schnirelman theory
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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