Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

Round 1

Reviewer 1 Report

The paper studied the Lie system on Lie group. It also proposed two families of methods based on Magnus expansions and on Runge-Kutta-Munthe-Kaas methods. Some examples are given to illustrated the conclusion. I have one problem:

Page 27,” the numerical schemes are based on geometric integrators, they inherit all the geometric properties”. The symmetry vectors are proposed in the paper. The corresponding conserved quantities should be obtained. The conserved quantities are the most important geometric properties for the Lie systems. The paper should discuss this point. That is, can the numerical simulation schemes preserve the conserved quantities?  

Minor editing of English language required

Author Response

Dear reviewer,

Thank you very much for your comment. Yes, the numerical schemes conserve the geometric properties. In these examples there are no interesting invariants that one can study, but right now we are preparing another application of our schemes to some manifolds that are not isomorphic to R^n, so we can see how certain geometric properties are conserved. Some of these examples are systems with curvature and topological invariants. We have added in the conclusions a couple of lines of how this could be addressed.

Thank you very much for your input.

Best, Cristina

Reviewer 2 Report

Please find my comments in the file attached.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

Thank you very much for your nice comments and review of our manuscript.

We have added a whole new subsection after 4.3 to clarify the numerical cost of the schemes as n rises. I hope it is more clear through the added examples.

We have also included the correction of all of the typos that you have pointed out and we have clarified the last sentence.

Thank you again for your input and I hope that now the manuscript is ready according to your criteria.

Reviewer 3 Report

The paper develops numerical methods for solving Lie systems, nonautonomous system of first-order ordinary differential equations with an autonomous superposition rule, adapted to the Lie structure of the system, unlike generic discretization methods. The methods are applied to several examples, including a linear quadratic optimal control problem. The method is of interest and adequately described and justified. There are some minor exposition problems, and multiple issues with English phrasing. I only list few of them, there are too many, the paper needs thorough proofreading by an English speaker. I recommend acceptance after minor revisions that address that.

See attached file for specifics.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

Thank you very much for all the suggestions and changes. We believe that your help has greatly enhanced our manuscript.

We have made all the English suggestions and concerning the theoretical questions:

About X_t, we have not changed it for X_T, since it is the very usual notation used in all of our papers concerning Lie systems. 

About Magnus method and Magnus methods, we use the plural because we have two different methods we focus on. Also, in the literature of Iserles they refer to them as plural, so we kept this way of referencing to them.

About the accuracy of nongeometric methods, we have certainly made some comment about it. Please find a little paragraph place at the very bottom of the conclusions.

We would like to thank you again for all of your very helpful and insightful input.