Geodesic boundary value problems with symmetry - Mathematics > Optimization and ControlReport as inadecuate

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Abstract: This paper shows how left and right actions of Lie groups on a manifold maybe used to complement one another in a variational reformulation of optimalcontrol problems equivalently as geodesic boundary value problems withsymmetry. We prove an equivalence theorem to this effect and illustrate it withseveral examples. In finite-dimensions, we discuss geodesic flows on the Liegroups SO3 and SE3 under the left and right actions of their respective Liealgebras. In an infinite-dimensional example, we discuss optimallarge-deformation matching of one closed curve to another embedded in the sameplane. In the curve-matching example, the manifold $\EmbS^1, \mathbb{R}^2$comprises the space of closed curves $S^1$ embedded in the plane$\mathbb{R}^2$. The diffeomorphic left action $\Diff\mathbb{R}^2$ deforms thecurve by a smooth invertible time-dependent transformation of the coordinatesystem in which it is embedded, while leaving the parameterisation of the curveinvariant. The diffeomorphic right action $\DiffS^1$ corresponds to a smoothinvertible reparameterisation of the $S^1$ domain coordinates of the curve. Aswe show, this right action unlocks an important degree of freedom forgeodesically matching the curve shapes using an equivalent fixed boundary valueproblem, without being constrained to match corresponding points along thetemplate and target curves at the endpoint in time.

Author: C. J. Cotter, D. D. Holm



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