Oh I see!
Unfortunately changing the parameters of this control (start and end date) in the Analysis section does not allow to have default values.
What you could do is to publish this analysis as a dashboard, making sure to expand the Advanced publish options and tick the Enable ad hoc filtering as in the picture below
In this way changing the values of start and end date will only temporarily affect the visuals and by refreshing the page you can get back the default values set in the filter definition of the analysis.
I hope this helps 