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 \({{\mathcal A}}\) of so-called prior vector fields. Because the physical laws are not known precisely, the measurements might not be 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. This is done by solving a variational problem, yielding so-called conditional extrema which satisfy an Euler–Lagrange equation. Then conservative prior fields on simply-connected Riemannian manifolds are characterised in terms of their conditional extrema. For specific prior fields on space forms, conditional extrema are obtained in terms of the Weierstrass elliptic function. Another class of examples comes from left-invariant prior fields on bi-invariant Lie groups, whose conditional extrema are shown to be right translations of pointwise-products of 1-parameter subgroups.
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Noakes, L. Conditional extremals. Math. Control Signals Syst. 24, 295–320 (2012). https://doi.org/10.1007/s00498-012-0081-3
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DOI: https://doi.org/10.1007/s00498-012-0081-3