Review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 24 November 2022

The manuscript describes the mathematical aspects of fractional calculus. Here are some remarks.

• Why are Monch and Mddotoch written in italics?

• What is MNC?

• Use brackets in SDEs after introducing its abbreviation.

• e.g. should be followed by a comma, not a semicolon.

• (??) to advance

• What is HF?

• The symbols in (1) and (2) are hard to read. Please use simpler ones.

• It is unclear what is new in comparison to the existing literature. The authors outlined the paper at the end of the introduction, but the readers are not sure which parts are new and which parts already existed in the body of published literature. Perhaps an additional bridging paragraph before this paragraph might help many readers to comprehend the idea of the paper.

• What is Tr? Is it trace?

• It was unclear why the authors shifted toward stochastic differential equations while in the beginning, they have never talked about it.

• What is HFD?

• The labels in Figures 1 and 2 are hardly visible and were way too small.

• Both Figures 1 and 2 do not have caption information.

• hasn't --> has not

• Example 2 outlined in Section 4.2 does not seem to connect with the message delivered in the paper. Again, one paragraph or two might help readers from mathematics to this particular application.

Second review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 9 December 2022

Thank you for revising the manuscript. Here are some minor remarks.

• Lines 23, 28: umlaut o should not be written in italics.

• Line 24: space needed.

• Pages 8, 11: "Tr" should not be in math mode/italics, it should be written in upright/normal mode.

• Perhaps a citation or two on the Lebesgue-dominated convergence theorem would be helpful to many readers.

• Page 9: Please confirm that the prime in the calligraphic L indicates a derivative with respect to a particular variable, which is rather unclear in this case what the variable is. Furthermore, aren't we talking about fractional derivatives? In that case, then the symbol might be confused with the regular/classical derivative with integer order.

• Is Bu a combined symbol, or are they separate symbols B and u?

• The term "equicontinuous" has never been defined. Its first appearance is in Lemma 7 but no explanation of what it constitutes.

• Line 175: Why does HF italics?

• Line 176: sin is not in italics mode.

• The labeling in Figure 1 is too small, and the quality is also poor. Please modify Figure 1 with a better-quality one.

• There is a mixture of British and American spellings.

• English is still poor and should be improved significantly. Here is an example of improved paragraphs. I leave it to the authors to find out what kind of changes I have made.

Digital filters perform an incredibly considerable aspect in the field of digital signal processing (DSP). Execution of digital filters is extraordinary; each of the essential factors that DSP has grown highly acclaimed. Commonly, we classify filters with two main usages: One is signal separation, and the other is signal restoration. Digital filters are an important entity in various fields of signal processing and unexceptionally received high-level prominence in the field of biomedical signal processing too. As most biomedical signals are low frequency by nature the general problem in those signals is to process the small amplitude of the signal naturally dominated by artifacts and various other noises. The efficiency of the system lies in retrieving those signals in a substantial amount of time to as not to generate a high level of delay in processing and analyzing those signals. In general, biomedical systems consist of complex cascaded blocks such that the delay in producing the output of the block cumulatively affects the generation of the final output. The speed, as well as accuracy of processing, is highly essential in modeling any biomedical system to retrieve such error-free processed signals.

Further, the biological signals are the one with minimal amplitudes and lower frequencies (few brain signal ranges from 1- 4Hz). The entire requirement of an error-free signal generation shall be attributed to the proper controllability of the modeled system obtained through the solution. The signal processing unit is implemented through a block diagram as depicted in Figure 1 using MATLAB-Simulink software with the aid of the proposed system of equations (1--2) shall be suitable for a few biomedical applications. The frequency response of the digital filter implemented using a fractional differentiation model as shown in Figure 2 exhibits sharp cut-off frequency with better signal restoration capability [46,47]. Therefore, the output is obtained, which is bounded and controllable with better stability in the amplitude in the low-frequency corner.

Motivated by the filter systems defined in [48–50], we present our filter system in Figure 1 and Figure 2. Figure 1 describes the rough pattern of the digital filter model, and Figure 2 describes the frequency response of the digital filter, which aids in enhancing the usefulness of the solution with a minimum measure of inputs.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 24 November 2022

The manuscript describes the mathematical aspects of fractional calculus. Here are some remarks.

• Why are Monch and Mddotoch written in italics?

• What is MNC?

• Use brackets in SDEs after introducing its abbreviation.

• e.g. should be followed by a comma, not a semicolon.

• (??) to advance

• What is HF?

• The symbols in (1) and (2) are hard to read. Please use simpler ones.

• It is unclear what is new in comparison to the existing literature. The authors outlined the paper at the end of the introduction, but the readers are not sure which parts are new and which parts already existed in the body of published literature. Perhaps an additional bridging paragraph before this paragraph might help many readers to comprehend the idea of the paper.

• What is Tr? Is it trace?

• It was unclear why the authors shifted toward stochastic differential equations while in the beginning, they have never talked about it.

• What is HFD?

• The labels in Figures 1 and 2 are hardly visible and were way too small.

• Both Figures 1 and 2 do not have caption information.

• hasn't --> has not

• Example 2 outlined in Section 4.2 does not seem to connect with the message delivered in the paper. Again, one paragraph or two might help readers from mathematics to this particular application.

Third review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 17 December 2022

• I think the mixture of British and American spellings has not been clearly resolved. Some grammatical mistakes persist.

• Lines 20-21: Reword the sentence starting with "The researchers Wang and Zhou ...".

• Line 29: "new discoveries" might be redundant.

• Line 55: remove "in".

• Line 55: Reword the sentence after that.

• Page 3: The symbols ... denote ... (wrong English).

• Line 157: missing a comma.

• Step:1 --> Step 1. (wrong colon location).

• Page 13: (again wrong English grammar) a ... Brownian motion ...

• The quality of Figure 1 is still poor and the labels are too small to read. (This previous remark has been ignored by the authors.)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Second review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 9 December 2022

Thank you for revising the manuscript. Here are some minor remarks.

• Lines 23, 28: umlaut o should not be written in italics.

• Line 24: space needed.

• Pages 8, 11: "Tr" should not be in math mode/italics, it should be written in upright/normal mode.

• Perhaps a citation or two on the Lebesgue-dominated convergence theorem would be helpful to many readers.

• Page 9: Please confirm that the prime in the calligraphic L indicates a derivative with respect to a particular variable, which is rather unclear in this case what the variable is. Furthermore, aren't we talking about fractional derivatives? In that case, then the symbol might be confused with the regular/classical derivative with integer order.

• Is Bu a combined symbol, or are they separate symbols B and u?

• The term "equicontinuous" has never been defined. Its first appearance is in Lemma 7 but no explanation of what it constitutes.

• Line 175: Why does HF italics?

• Line 176: sin is not in italics mode.

• The labeling in Figure 1 is too small, and the quality is also poor. Please modify Figure 1 with a better-quality one.

• There is a mixture of British and American spellings.

• English is still poor and should be improved significantly. Here is an example of improved paragraphs. I leave it to the authors to find out what kind of changes I have made.

Digital filters perform an incredibly considerable aspect in the field of digital signal processing (DSP). Execution of digital filters is extraordinary; each of the essential factors that DSP has grown highly acclaimed. Commonly, we classify filters with two main usages: One is signal separation, and the other is signal restoration. Digital filters are an important entity in various fields of signal processing and unexceptionally received high-level prominence in the field of biomedical signal processing too. As most biomedical signals are low frequency by nature the general problem in those signals is to process the small amplitude of the signal naturally dominated by artifacts and various other noises. The efficiency of the system lies in retrieving those signals in a substantial amount of time to as not to generate a high level of delay in processing and analyzing those signals. In general, biomedical systems consist of complex cascaded blocks such that the delay in producing the output of the block cumulatively affects the generation of the final output. The speed, as well as accuracy of processing, is highly essential in modeling any biomedical system to retrieve such error-free processed signals.

Further, the biological signals are the one with minimal amplitudes and lower frequencies (few brain signal ranges from 1- 4Hz). The entire requirement of an error-free signal generation shall be attributed to the proper controllability of the modeled system obtained through the solution. The signal processing unit is implemented through a block diagram as depicted in Figure 1 using MATLAB-Simulink software with the aid of the proposed system of equations (1--2) shall be suitable for a few biomedical applications. The frequency response of the digital filter implemented using a fractional differentiation model as shown in Figure 2 exhibits sharp cut-off frequency with better signal restoration capability [46,47]. Therefore, the output is obtained, which is bounded and controllable with better stability in the amplitude in the low-frequency corner.

Motivated by the filter systems defined in [48–50], we present our filter system in Figure 1 and Figure 2. Figure 1 describes the rough pattern of the digital filter model, and Figure 2 describes the frequency response of the digital filter, which aids in enhancing the usefulness of the solution with a minimum measure of inputs.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 24 November 2022

The manuscript describes the mathematical aspects of fractional calculus. Here are some remarks.

• Why are Monch and Mddotoch written in italics?

• What is MNC?

• Use brackets in SDEs after introducing its abbreviation.

• e.g. should be followed by a comma, not a semicolon.

• (??) to advance

• What is HF?

• The symbols in (1) and (2) are hard to read. Please use simpler ones.

• It is unclear what is new in comparison to the existing literature. The authors outlined the paper at the end of the introduction, but the readers are not sure which parts are new and which parts already existed in the body of published literature. Perhaps an additional bridging paragraph before this paragraph might help many readers to comprehend the idea of the paper.

• What is Tr? Is it trace?

• It was unclear why the authors shifted toward stochastic differential equations while in the beginning, they have never talked about it.

• What is HFD?

• The labels in Figures 1 and 2 are hardly visible and were way too small.

• Both Figures 1 and 2 do not have caption information.

• hasn't --> has not

• Example 2 outlined in Section 4.2 does not seem to connect with the message delivered in the paper. Again, one paragraph or two might help readers from mathematics to this particular application.

Fourth review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 20 December 2022

• The caption in both figures are still minimum and needs further explanation.

• Even though the figure has been changed, it is still a low quality.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Third review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 17 December 2022

• I think the mixture of British and American spellings has not been clearly resolved. Some grammatical mistakes persist.

• Lines 20-21: Reword the sentence starting with "The researchers Wang and Zhou ...".

• Line 29: "new discoveries" might be redundant.

• Line 55: remove "in".

• Line 55: Reword the sentence after that.

• Page 3: The symbols ... denote ... (wrong English).

• Line 157: missing a comma.

• Step:1 --> Step 1. (wrong colon location).

• Page 13: (again wrong English grammar) a ... Brownian motion ...

• The quality of Figure 1 is still poor and the labels are too small to read. (This previous remark has been ignored by the authors.)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Second review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 9 December 2022

Thank you for revising the manuscript. Here are some minor remarks.

• Lines 23, 28: umlaut o should not be written in italics.

• Line 24: space needed.

• Pages 8, 11: "Tr" should not be in math mode/italics, it should be written in upright/normal mode.

• Perhaps a citation or two on the Lebesgue-dominated convergence theorem would be helpful to many readers.

• Page 9: Please confirm that the prime in the calligraphic L indicates a derivative with respect to a particular variable, which is rather unclear in this case what the variable is. Furthermore, aren't we talking about fractional derivatives? In that case, then the symbol might be confused with the regular/classical derivative with integer order.

• Is Bu a combined symbol, or are they separate symbols B and u?

• The term "equicontinuous" has never been defined. Its first appearance is in Lemma 7 but no explanation of what it constitutes.

• Line 175: Why does HF italics?

• Line 176: sin is not in italics mode.

• The labeling in Figure 1 is too small, and the quality is also poor. Please modify Figure 1 with a better-quality one.

• There is a mixture of British and American spellings.

• English is still poor and should be improved significantly. Here is an example of improved paragraphs. I leave it to the authors to find out what kind of changes I have made.

Digital filters perform an incredibly considerable aspect in the field of digital signal processing (DSP). Execution of digital filters is extraordinary; each of the essential factors that DSP has grown highly acclaimed. Commonly, we classify filters with two main usages: One is signal separation, and the other is signal restoration. Digital filters are an important entity in various fields of signal processing and unexceptionally received high-level prominence in the field of biomedical signal processing too. As most biomedical signals are low frequency by nature the general problem in those signals is to process the small amplitude of the signal naturally dominated by artifacts and various other noises. The efficiency of the system lies in retrieving those signals in a substantial amount of time to as not to generate a high level of delay in processing and analyzing those signals. In general, biomedical systems consist of complex cascaded blocks such that the delay in producing the output of the block cumulatively affects the generation of the final output. The speed, as well as accuracy of processing, is highly essential in modeling any biomedical system to retrieve such error-free processed signals.

Further, the biological signals are the one with minimal amplitudes and lower frequencies (few brain signal ranges from 1- 4Hz). The entire requirement of an error-free signal generation shall be attributed to the proper controllability of the modeled system obtained through the solution. The signal processing unit is implemented through a block diagram as depicted in Figure 1 using MATLAB-Simulink software with the aid of the proposed system of equations (1--2) shall be suitable for a few biomedical applications. The frequency response of the digital filter implemented using a fractional differentiation model as shown in Figure 2 exhibits sharp cut-off frequency with better signal restoration capability [46,47]. Therefore, the output is obtained, which is bounded and controllable with better stability in the amplitude in the low-frequency corner.

Motivated by the filter systems defined in [48–50], we present our filter system in Figure 1 and Figure 2. Figure 1 describes the rough pattern of the digital filter model, and Figure 2 describes the frequency response of the digital filter, which aids in enhancing the usefulness of the solution with a minimum measure of inputs.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Review Manuscript ID: mathematics-2074391 Type of manuscript: Article Title: A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators Authors: S. Sivasankar, R. Udhayakumar *, Muchenedi Hari Kishor, Sharifah E. Al Hazmi, S. Al-Omari Submitted to the section: Dynamical Systems, Fractional Calculus, and Mathematical Applications Date: 24 November 2022

The manuscript describes the mathematical aspects of fractional calculus. Here are some remarks.

• Why are Monch and Mddotoch written in italics?

• What is MNC?

• Use brackets in SDEs after introducing its abbreviation.

• e.g. should be followed by a comma, not a semicolon.

• (??) to advance

• What is HF?

• The symbols in (1) and (2) are hard to read. Please use simpler ones.

• It is unclear what is new in comparison to the existing literature. The authors outlined the paper at the end of the introduction, but the readers are not sure which parts are new and which parts already existed in the body of published literature. Perhaps an additional bridging paragraph before this paragraph might help many readers to comprehend the idea of the paper.

• What is Tr? Is it trace?

• It was unclear why the authors shifted toward stochastic differential equations while in the beginning, they have never talked about it.

• What is HFD?

• The labels in Figures 1 and 2 are hardly visible and were way too small.

• Both Figures 1 and 2 do not have caption information.

• hasn't --> has not

• Example 2 outlined in Section 4.2 does not seem to connect with the message delivered in the paper. Again, one paragraph or two might help readers from mathematics to this particular application.