Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

2011 ◽  
Vol 202 (11) ◽  
pp. 1593-1615 ◽  
Author(s):  
Andrei A Ardentov ◽  
Yurii L Sachkov
Keyword(s):  
Engel Group ◽  
Extremal Trajectories

Extremal trajectories in the sub-Lorentzian problem on the Engel group

2018 ◽  
Vol 209 (11) ◽  
pp. 1547-1574
Author(s):  
A. A. Ardentov ◽  
Yu. L. Sachkov ◽  
T. Huang ◽  
X. Yang
Keyword(s):  
Engel Group ◽  
Extremal Trajectories

On 4-Engel Groups

2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Normal Closure ◽  
Engel Group

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group

scholarly journals Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

2017 ◽  
Vol 22 (8) ◽  
pp. 909-936 ◽  
Author(s):  
Andrei A. Ardentov ◽  
Yuri L. Sachkov
Keyword(s):  
Cut Locus ◽  
Engel Group ◽  
Maxwell Strata

Control Systems on the Engel Group

2018 ◽  
Vol 25 (3) ◽  
pp. 377-402
Author(s):  
D. I. Barrett ◽  
C. E. McLean ◽  
C. C. Remsing
Keyword(s):  
Control Systems ◽  
Engel Group

scholarly journals Path planning in a Riemannian manifold using optimal control

2020 ◽  
Vol 17 (12) ◽  
pp. 2050181
Author(s):  
Souma Mazumdar
Keyword(s):  
Optimal Control ◽  
Riemannian Manifold ◽  
Initial Point ◽  
Geodesic Curvature ◽  
Optimal Controls ◽  
Arc Length ◽  
Final Point ◽  
Minimization Principle ◽  
Pontryagin Principle ◽  
Extremal Trajectories

We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time [Formula: see text] and shown to have norm [Formula: see text] which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.

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