Bibliography style with only the initials of the first names

You could proceed as follows to direct BibTeX to abbreviate first (and middle) names down to the initials, while using a suitably modified version of the plain bibliography style:

• Find the file plain.bst in your TeX distribution. (One way to find this file is to issue the command kpsewhich plain.bst at a command prompt.) Copy this file to, say myplain.bst. (Don't edit directly a file provided as part of your TeX distribution.)

• Open the file myplain.bst in your favorite text editor -- the one you use to edit your .tex files will do -- and search for the function named format.names. (In the version of this file that's in my TeX distribution, the function format.names starts on line 185.)

• In this function, look for the line

  { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't :=
  

  Change this line to

  { s nameptr "{f. }{vv~}{ll}{, jj}" format.name$ 't :=
  

• Save the file myplain.bst either to the directory that contains your main .tex file or to a directory that's searched by your BibTeX distribution. (If you choose the second option, depending on your TeX distribution, you may also have to refresh the filename database of the TeX distribution.)

• Start using the modified bibliography style file by issuing the command \bibliographystyle{myplain}, and be sure to run LaTeX, BibTeX, and LaTeX twice more on the main .tex file to propagate all changes.

Here's how the entry in question will be typeset with myplain as the bibliography style:

\RequirePackage{filecontents}
\documentclass{article}
\begin{filecontents*}{\jobname.bib}
@article{Farrell:2005aa,
    Author = {Farrell, Jay and Sharma, Manu and Polycarpou, Marios},
    Cited-By = {113},
    Date-Added = {2014-03-11 16:36:51 +0000},
    Date-Modified = {2014-03-11 16:38:05 +0000},
    Journal = {Journal of Guidance, Control, and Dynamics},
    Number = {6},
    Pages = {1089-1102},
    Title = {Backstepping-based flight control with adaptive 
            function approximation},
    Volume = {28},
    Year = {2005},
}
\end{filecontents*}
\bibliographystyle{myplain} % note: 'myplain' rather than 'plain'
\begin{document}
\nocite{*}
\bibliography{\jobname}
\end{document}

You can use biblatex with convenient options. Write:

 ......................   
\usepackage[style=numeric,firstinits]{biblatex}
\addbibresource{MyBibfile.bib}
 ....................
\begin{document}
 ....................
\printbibliography
\end{document}

This makes both citation and bibliography styles numeric. But you can choose citation style and bibliography style independently. Take a look at §3.3 of the biblatex documentation for a description of the standard syles available. Moreover, in case you use natbib, you can load biblatex with the natbib option, that emulates natbib functionalities.

Updated (3/9/2017)

The new syntax for first names initials is [giveninits].