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Abstract
This paper shows how commuting left and right actions of Lie groups
on a manifold may be used to complement one another in a variational
reformulation of optimal control problems as geodesic boundary value
problems with symmetry. In such problems, the endpoint boundary
condition is only specified up to the right action of a symmetry
group. In this paper we show how to reformulate the problem by
introducing extra degrees of freedom so that the endpoint condition
specifies a single point on the manifold. We prove an equivalence
theorem to this effect and illustrate it with several examples. In
finite-dimensions, we discuss geodesic flows on the Lie groups
$SO(3)$ and $SE(3)$ under the left and right actions of their
respective Lie algebras. In an infinite-dimensional example, we
discuss optimal large-deformation matching of one closed oriented
curve to another embedded in the same plane. In the curve-matching
example, the manifold Emb$(S^1, \mathbb{R}^2)$ comprises the space
$S^1$ embedded in the plane $\mathbb{R}^2$. The diffeomorphic left
action Diff$(\mathbb{R}^2)$ deforms the curve by a smooth
invertible time-dependent transformation of the coordinate system in
which it is embedded, while leaving the parameterisation of the
curve invariant. The diffeomorphic right action Diff$(S^1)$
corresponds to a smooth invertible reparameterisation of the $S^1$
domain coordinates of the curve. As we show, this right action
unlocks an important degree of freedom for geodesically matching the
curve shapes using an equivalent fixed boundary value problem,
without being constrained to match corresponding points along the
template and target curves at the endpoint in time.
Mathematics Subject Classification: Primary: 70H33, 58E50; Secondary: 92C55.
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