RadSplitButton provides a flexible client-side API. You can easily interact with the button in the browser using the RadSplitButton client-side object. In addition to a variety of client-side events, the client-side object model lets you achieve complicated tasks while avoiding unnecessary post-backs.
var button = $find("<%= RadSplitButton.ClientID %>");
Check out the Get Client-side Reference to a Control Object article to find other options to reach the control object on the client-side.