Fundamental solutions for two-point boundary value problems in orbital mechanics
type: Item
instance of: scholarly article
title: Fundamental solutions for two-point boundary value problems in orbital mechanics
zbMATH Open document ID: 1386.49054
author: Seung Hak Han
author: William M. Mceneaney
published in: Applied Mathematics and Optimization
publication date: 2018-03-14T00:00:00Z
review text: The paper by Seung Hak Han and William M. McEneaney contributes in a valuable way to the mathematical interplay between Optimal Control Theory and the Theory of Differential Games on an exciting field of motivation and application, namely, orbital mechanics, that includes both (i) astronomy, and (ii) aerospace engineering. This work is rigorous, and based on it, future research may be established and very much emerging real-life applications. Indeed, the authors are concerned with a Two-Point Boundary Value Problem (TPBVP) in orbital mechanics, with a rather body and a number of larger bodies. The small body may, e.g., be a spacecraft or an asteroid. The authors convert the Least-Action Principle TPBVP formulation into an initial value problem through the addition of an appropriate terminal cost to the objective or ``action'' functional. That formulation is then taken to get a fundamental solution, which can be used to resolve the TPBVP for a family of boundary conditions within a particular class. Especially, Convex Duality allows for interpreting the least-action principle as a Differential Game, in which an opposing agent maximizes subject to an indexed set of quadratics to receive the gravitational potential. If the time duration is smaller than a particular bound, then there is a unique critical point for the resulting differential game, yielding the fundamental solution in terms of the solutions of Riccati equations associated. This excellent work is timely and well-embedded into the contemporary research landscape, structured and written well, deep, exemplified and illustrated numerically well. The seven sections of this rich work are these: 1. Introduction, 2. Problem Statement and Fundamental Solution, 3. Optimal Control Problem, 4. Differential Game Formulation, with the subsections 4.1. Revisiting the Payoff, 4.2. Existence and Uniqueness of Optimal, Controls: \(c<\infty\), 4.3. Existence and Uniqueness of Optimal Controls: \(c=\infty\), and 4.4. Hamilton…
reviewed by: Gerhard-Wilhelm Weber
Mathematics Subject Classification ID: 49N70
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 35D40
Mathematics Subject Classification ID: 70M20
zbMATH DE Number: 6850156
zbMATH Keywords: least action
zbMATH Keywords: two-point boundary value problem
zbMATH Keywords: differential game
zbMATH Keywords: Hamilton-Jacobi
zbMATH Keywords: optimal control
MaRDI profile type: Publication
full work available at URL: s00245-016-9369-x
OpenAlex ID: W2475102454
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DOI: 10.1007/S00245-016-9369-X
description: scientific article; zbMATH DE number 6850156