Two-dimensional almost-Riemannian structures with tangency points

Agrachev, Andrei; Boscain, Ugo; Charlot, G.; Ghezzi, Roberta; Sigalotti, Mario

doi:10.1016/j.anihpc.2009.11.011

Cited by 59 publications

(58 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…• singularity theory, providing a starting point to the classification of extremals (see [136,137,138,19,41,139]) et permitting to study how trajectories may lose optimality (see [140,141,142]);…”

Section: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

View full text Add to dashboard Cite

This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

show abstract

Section: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

View full text Add to dashboard Cite

show abstract

“…The singular set Z ⊂ N is the set of points where dim D q < n. Tagency points have deep consequences on the local structure of the almost-Riemannian metric structure, and have been studied, in the 2-dimensional case, in [3,7]. If Z is a smooth, embedded submanifold, for all q ∈ Z there exists a non-zero λ ∈ T * q N , defined up to multiplication by a constant, such that λ(T q Z) = 0.…”

Section: Almost-riemannian Geometrymentioning

confidence: 99%

Quantum confinement on non-complete Riemannian manifolds

2018

View full text Add to dashboard Cite

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure ω, possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V ∈ L 2 loc (M ). The main merit of this paper is the identification of an intrinsic quantity, the effective potential V eff , which allows to formulate simple criteria for quantum confinement. Let δ be the distance from the possibly non-compact metric boundary of M . A simplified version of the main result guarantees quantum completeness if V ≥ −cδ 2 far from the metric boundary andclose to the metric boundary.These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of M ; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [9].

show abstract

“…This geometry goes back to [13] and [21]. It appears as a part of sub-Riemannian geometry, and has aroused some interest, as shown by the recent papers [2], [3], [7], [8], [9], [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Isometries of almost-Riemannian structures on Lie groups

Jouan

Zsigmond²,

Ayala³

2018

Differential Geometry and its Applications

View full text Add to dashboard Cite

A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) − 1 left-invariant ones.It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism.These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.

show abstract

Two-dimensional almost-Riemannian structures with tangency points

Cited by 59 publications

References 13 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Quantum confinement on non-complete Riemannian manifolds

Isometries of almost-Riemannian structures on Lie groups

Contact Info

Product

Resources

About