To serve our visitors, I want thejm to be able to search the page they are on and not search the entire website. Is this possible? The sesarch control does not have an option to search either the entire site of the current page.
It wouldn't be possible to configure out own search element to search a specific page, unfortunately. You might be able to use a third party search widget for something like that, e.g. Google Site Search. I'm not sure if it's possible with that in particular, but it's worth checking out since you could probably include it with an Embed Code element.