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-Jacobi-Bellman PDE, 5. The Fundamental Solution in Terms of Riccati Equation Solutions, 6. The Maximization over \(\alpha\) [of the solution of Riccati equations with respect to the maximizing player], 6.1. Derivative of [the solution] with Respect to \(\alpha\), 6.2. An Approximate Solution, 6.3. Error Analysis, and 6.4. First-Order Necessary Condition for Maximization of [functional], 7. Examples, with the subsections 7.1. Specifics of Two Problem Classes, 7.2. Example 1: [equality condition], and 7.3. Example 2: [linear equality condition].
Future refinements in theory and numerical methods could be expected by the research community, inspired by this strong paper. Those might even be made in terms of Stochastic Optimal Control with Jumps, of Stochastic Game Theory, etc., addressing further fields of motivation and application, too.
Such advances could foster progress in science and technology, astronomy, environmental and earth-sciences, finance and economics, bio- and neurosciences, social and developmental sciences.