Dynamic programming solution to minimal surface of revolution (Q2762827)

The title is a good description of the subject: the Euler-Plateau problem for minimal surfaces of revolution, considered from the dynamic point of view. Using some recent results, the author gives a complete and rigorous solution of this problem, shorter than those given previously by other authors and eliminating many of the intuitively geometric considerations. The high mathematical level and the difficult framework prevent us to give an accurate description of the results, and hence we only expose the principal stages of proofs and outline the basic methods. The main theorem establishes that for a particular \(C^1\) mapping \(X^{(1)}(\cdot,\cdot)\), there exists exactly two \(C^1\) stratified inverses, and takes about two pages of proof. NEWLINENEWLINENEWLINEThe paper, consisting in three sections and an introduction, presents the dynamic programming method for general Bolza autonomous optimal control problem, states the dynamic programming algorithm for autonomous optimal control problem and the solution of the minimal revolution surface problem. The author's point of view has some advantages, when compared with other authors' points of view: 1. the possibility to treat the complicated problems with phase-space constraints using the Goldschmidt's trajectories; 2. the existence of stratified value functions in optimal control theory; 3. the possibility to decide the optimality in the case of several extremal trajectories passing through the same point; 4. the distinction between proper and apparent restrictions of optimal control problems. NEWLINENEWLINENEWLINEThe author handles with ease complicated techniques and uses the most recent results in the field, many of them being his own contributions.