About simple variational splines from the Hamiltonian viewpoint

Abstract

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

45 pages, 8 figures

References in corpus (11)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Lagrangian submanifolds and dynamics on Lie algebroids
• Sobolev metrics on shape space of surfaces
• Invariant higher-order variational problems
• Statistical analysis on high-dimensional spheres and shape spaces
• Invariant higher-order variational problems II
• Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
• Cubic polynomials on Lie groups: reduction of the Hamiltonian system
• Inexact trajectory planning and inverse problems in the Hamilton--Pontryagin framework
• Geometric integrators for higher-order mechanics on Lie groups
• Optimal control of affine connection control systems from the point of view of Lie algebroids