Geodesic boundary value problems with symmetry

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.