Name
Title | ||
|---|---|---|
Existence, Multiplicity, and Regularity for sub-Riemannian Geodesics by Variational Methods |
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. |
Year | DOI | Venue |
|---|---|---|
2002 | 10.1109/CDC.2002.1184808 | Siam Journal on Control and Optimization |
Keywords | Field | DocType |
riemannian manifold,smooth immersion,schnirelman theory,continuous curves horizontal,sub-riemannian manifold,sub-riemannian geodesics,integro-differential equation,ordinary differential equation,variational methods,variational theory,smooth distribution,riemannian energy,differential geometry,absolute continuity,variational method,exponential map,integro differential equation | Codimension,Mathematical optimization,Ordinary differential equation,Riemannian manifold,Submanifold,Differential geometry,Exponential map (Riemannian geometry),Geodesic,Manifold,Mathematics | Journal |
Volume | Issue | ISSN |
40 | 6 | 0363-0129 |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Roberto Giambò | 1 | 11 | 3.41 |
| Fabio Giannoni | 2 | 8 | 2.66 |
| Paolo Piccione | 3 | 5 | 1.25 |