Conditional Extremals

TL;DR

This paper investigates the mathematical properties of conditional extrema, which are optimal interpolations of system trajectories given uncertain prior laws, with explicit solutions in special geometric cases.

Contribution

It provides a foundational analysis of conditional extrema, characterizing their properties under various prior assumptions and deriving explicit solutions in symmetric and Lie group settings.

Findings

Conditional extrema generalize geodesics when prior fields are non-zero.

Explicit solutions involve elliptic functions for symmetric priors.

On Lie groups, conditional extrema relate to 1-parameter subgroups.

Abstract

Imagine that measurements are made at times $t_0$ and $t_1$ of the trajectory of a physical system whose governing laws are given approximately by a class ${\cal A}$ of so-called {\em prior vector fields}. Because the physical laws are not known precisely, it might be that the measurements are not realised by the integral curve of any prior field. We want to estimate the behaviour of the physical system between times $t_0$ and $t_1$. An integral curve of an arbitrary vector field $X$ is said to be {\em feasible} when it interpolates the measurements. When $X$ is critical for $L^2$ distance to ${\cal A}$, the feasible curve is called a {\em conditional extremum}. When the distance to ${\cal A}$ is actually minimal, the conditional extremum is a best estimate for the intermediate behaviour of the system. The present paper does some of basic groundwork for investigating mathematical properties of conditional extrema, focusing on cases where ${\cal A}$ comprises a single prior field $A$. When $A={\bf 0}$ a conditional extremal is a geodesic arc, but this special case is not very representative. In general, $A$ enters into the Euler-Lagrange equation for conditional extrema, and more can be said when $A$ is conservative or has special symmetry. We characterise conservative priors on simply-connected Riemannian manifolds in terms of their conditional extrema: when time is reversed, a constant is added to the $L^2$ distance. For some symmetric priors on space forms we obtain conditional extrema in terms of the Weierstrass elliptic function. For left-invariant priors on bi-invariant Lie groups, conditional extrema are shown to be right translations of pointwise-products of 1-parameter subgroups.