Application of the Averaging Method to the Optimal Control Problem of Non-Linear Differential Inclusions on the Finite Interval

Round 1

Reviewer 1 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

Point 1: The approximate interval is [0,T], can T be extend to a number with respect to the small parameter ε? 

Response 1: Such extension is possible. We plan to investigate it in our future research. 

Point 2: To make the paper self-consistent, the authors should state Lions’ Lemma in this paper. 

Response 2: We agree with the reviewer’s comment. We have rephrased its mention and added a proper reference (now lines 100, 136). 

Point 3: In order to make the results clear and easy to understand, the authors should give a specific example before the Conclusion part. 

Response 3: We have added an example just before the Conclusion part of the paper. 

Point 4: Some typos and English mistakes should be checked. 

Response 4: Thank you for your comment. We conducted typos and mistake checks and corrections. 

Reviewer 2 Report

In the present paper, authors consider the optimal control problem for a system of differential inclusions with fast oscillating in time parameters and coercive cost functional. In such problems, the common approach is to transit to averaged parameters which may significantly simplify the problem. The main result is to prove the approximate optimality of the optimal control of the problem with averaged coefficients for the initial perturbed one (Theorem 3 in the paper). The considered inclusions are assumed to be non-linear with respect to the control variable which brings additional technical problems and requires rather sophisticated arguments concerning the multi-valued  Krasnoselskiy-Krein Theorem.
In my opinion, the paper is well-written and all proofs are correct.
I recommend this paper for publication.
I have only a few comments and remarks:
1) I do not understand the sense of Remark 4.
2) I think that the notion of minimizing sequence in optimal control problems is quite clear, so the sentence between line 84 and line 85 can be dropped out.
3) Line 89: The Arzelà–Ascoli theorem  instead of the Arcel’s theorem
4) Line 99:  authors used some variant of the dominated convergence theorem which they mentioned as the Lions Lemma. The citation needs to be added here.
5) There are many papers devoted to close subjects (approximate control, robust control) in the mathematical literature. Therefore, the list of references should be completed by some recent results in this area.

Author Response

Point 1: I do not understand the sense of Remark 4. 

Response 1: Thank you for bringing this remark to our attention. Indeed, in the context of the research conducted in this paper, this remark is needless and it is dropped out (lines 74-75).

Point 2: I think that the notion of minimizing sequence in optimal control problems is quite clear, so the sentence between line 84 and line 85 can be dropped out.

Response 2: Thank you for the note. The sentence between (now) lines 85 and 86 is dropped out.

Point 3: Line 89: The Arzelà–Ascoli theorem  instead of the Arcel’s theorem. 

Response 3: We mention the Arzelà–Ascoli theorem instead of the Arcel’s theorem (now line 90). 

Point 4: Line 99:  authors used some variant of the dominated convergence theorem which they mentioned as the Lions Lemma. The citation needs to be added here. 

Response 4: We agree with the reviewer’s comment. We have rephrased its mention and added a proper reference (now lines 100, 136). 

Point 5: There are many papers devoted to close subjects (approximate control, robust control) in the mathematical literature. Therefore, the list of references should be completed by some recent results in this area. 

Response 5: Thank you for this comment. We updated the Introduction and added some papers where similar problems are considered (e.g. minimax, robust and adaptive control) to complete the list of references with recent results in the area (lines 7-8 and references [8-10]).

Round 2

Reviewer 1 Report

The authors have addressed all my issues. I recommend for acceptance.