ODE trajectories as abnormal curves in Carnot groups

Hakavuori, Eero

doi:10.1016/j.jde.2021.07.046

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…Despite 30 years of efforts, there are not many general results. Among the latter we must mention the derivation of the second-order necessary conditions for optimality [2] (known as Goh conditions) and a technical masterpiece of Hakavouri and Le Donne [8] (see also [9]) which ruled out corner-like singularities (some previous results in this direction are [14,19]). We are also aware of a very recent attempts to derive third-order (and higher) necessary conditions of optimality [4,5].…”

Section: Introductionmentioning

confidence: 99%

New second-order optimality conditions in sub-riemannian geometry

Jóźwikowski

2023

ESAIM: COCV

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We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions.

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Section: Introductionmentioning

confidence: 99%

New second-order optimality conditions in sub-riemannian geometry

Jóźwikowski

2023

ESAIM: COCV

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“…Also nice abnormal extremals, see [17], are locally length minimizing. Examples of purely Lipschitz and spiral-like abnormal curves in Carnot groups are presented in [14,15], and an algorithm for producing many new examples is proposed in [9]. The length-minimality property of all these examples is not yet well-understood.…”

Section: Introductionmentioning

confidence: 99%