10 research outputs found
A contact covariant approach to optimal control with applications to sub-Riemannian geometry
We discuss contact geometry naturally related with optimal control problems
(and Pontryagin Maximum Principle). We explore and expand the observations of
[Ohsawa, 2015], providing simple and elegant characterizations of normal and
abnormal sub-Riemannian extremals.Comment: A small correction in the statement and proof of Thm 6.15. Watch our
publication: https://youtu.be/V04N9X3NxYA and https://youtu.be/jghdRK2IaU
New second-order optimality conditions in sub-Riemannian Geometry
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. We also prove
that the assumptions weaker than these used in [Boarotto, Monti, Palmurella,
2020] to derive third-order Goh conditions, imply piece-wise- regularity
of an abnormal extremal.Comment: Vastly improved presentation and proof, to appear in ESAIM Control
Optim. Calc. Va