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Consider the following BibTeX entry (I think it was downloaded from ScienceDirect)

@article{busoniu2018,
    author = {Lucian Bu{\c s}oniu and Tim [de Bruin] and Domagoj Toli{\'c} and Jens Kober and Ivana Palunko},
    doi = {https://doi.org/10.1016/j.arcontrol.2018.09.005},
    issn = {1367-5788},
    journal = {Annual Reviews in Control},
    keywords = {Reinforcement learning, Optimal control, Deep learning, Stability, Function approximation, Adaptive dynamic programming},
    pages = {8 - 28},
    title = {Reinforcement learning for control: Performance, stability, and deep approximators},
    url = {http://www.sciencedirect.com/science/article/pii/S1367578818301184},
    volume = {46},
    year = {2018},
    bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S1367578818301184},
    bdsk-Url-2 = {https://doi.org/10.1016/j.arcontrol.2018.09.005},
}

The last name of the second author appears enclosed in square brackets. I kept having problems when citing this article until I changed those square brackets [] to curly braces {}.

I searched BibTeX docs, and could not find any mention to the use of [] in authors' names. Does its usage differ from using {}? Or is this probably a mistake from the website that generated the citation?

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    This kind of name needs a special treatment, more than just enclosing it in (any kind of) brackets. It is a Dutch name with a separate prefix that should be sorted under 'B' for 'Bruin' and not 'D' for 'de'. See for example texfaq.org/FAQ-bibprefixsort on how to do this with BibTeX. With BibLaTeX it is a bit easier, in that case you can use the useprefix=false option, see for example tex.stackexchange.com/questions/226893/….
    – Marijn
    Jul 7 '20 at 12:07
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[ and ] have no special meaning in name fields for BibTeX. { and } do have special meaning, so turning [de Bruin] into {de Bruin} would result in very different outcomes.

BibTeX parses Tim [de Bruin] as

  • last name Bruin],
  • name prefix/"von part" [de and
  • first name Tim.

This is pretty much the same result you would get from Tim de Bruin minus the brackets. Tim {de Bruin} on the other hand would be parsed as

  • last name {de Bruin} and
  • first name Tim

without a prefix.

The problems with the square brackets really start when a bibliography style tries to pass on data to LaTeX via the optional argument of \bibitem. The following MWE

\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{natbib}

\begin{filecontents}{\jobname.bib}
@article{busoniu2018,
  author  = {Lucian Bu{\c s}oniu and Tim [de Bruin]
             and Domagoj Toli{\'c} and Jens Kober
             and Ivana Palunko},
  doi     = {10.1016/j.arcontrol.2018.09.005},
  journal = {Annual Reviews in Control},
  pages   = {8-28},
  title   = {Reinforcement Learning for Control:
             Performance, Stability, and Deep Approximators},
  volume  = {46},
  year    = {2018},
}
\end{filecontents}

\begin{document}
\cite{busoniu2018}
\bibliographystyle{plainnat}
\bibliography{\jobname}
\end{document}

Produces the output

Tolić, Kober, and Palunko]busoniu2018 Lucian Buşoniu, Tim [de Bruin], Domagoj Tolić, Jens Kober, and Ivana Palunko. Reinforcement learning for control: Performance, stability, and deep approximators. Annual Reviews in Control, 46:8–28, 2018. doi: 10.1016/j.arcontrol.2018.09.005.

To understand why that happens we need to take a look at the .bbl file. The .bbl file contains the printable bibliography and is essentially just \input when you say \bibliography{...}. The relevant entry shows as

\bibitem[Bu{\c s}oniu et~al.(2018)Bu{\c s}oniu, [de Bruin], Toli{\'c}, Kober,
  and Palunko]{busoniu2018}
Lucian Bu{\c s}oniu, Tim [de Bruin], Domagoj Toli{\'c}, Jens Kober, and Ivana
  Palunko.
\newblock Reinforcement learning for control: Performance, stability, and deep
  approximators.
\newblock \emph{Annual Reviews in Control}, 46:\penalty0 8--28, 2018.
\newblock \doi{10.1016/j.arcontrol.2018.09.005}.

Every LaTeX user who has tried to cite with page reference in a theorem heading (as in the doomed *\begin{theorem}[\cite[Thm.~3.24]{sigfridsson}], see Bug? LaTeX misparses nested optional arguments) will spot the problem: The optional argument of \bibitem contains unescaped square brackets.

In particular the closing bracket in [de Bruin] terminates the optional argument early leading to chaos with the rest of the argument structure.

That explains why the brackets in Tim [de Bruin] are bad. It doesn't explain why they are there in the first place. They are not present in anything written on the article landing page https://doi.org/10.1016/j.arcontrol.2018.09.005. I could not get hold of the PDF version of the original article from Elsevier, but I found a manuscript version of the paper on the website of one of the authors and there is no indication of square brackets in that paper version.

I can only speculate and would probably come to a similar conclusion as Marijn in the comments. Dutch names with a prefix like "de", "van" follow a peculiar set of rules (last names are usually sorted without the prefix and there are certain capitalisation requirements; to add insult to injury the rules differ between the Netherlands and Belgium, so if you want to be sure to treat someone's name correctly, you need more details). Presumably someone thought that the square brackets would help BibTeX to parse the name better (they don't, as we've seen) or to indicate to people downloading the file that the name needs special attention (that might have worked: you noticed a problem).

The issue with sorting "de Bruin" is probably not much of an issue with this particular example, since he is the second author, so a problem would only arise if there were another work with first author Buşoniu in your document. But in case you want things done correctly according to Dutch rules you probably need some manual intervention, see https://texfaq.org/FAQ-bibprefixsort.

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