I tried to find a nice solution to this with Date::Manip, but I couldn't. There probably is a nicer solution, so please tell me. Anyway, this should work, I hope, even if it's ugly:

use warnings;
use Date::Manip 6.30; 
my $today = Date::Manip::Date->new(q(today)); 
my $sunday = $today->new($today->calc($today->new_delta("+1day"))
->printf("%G-W%W-1"))->calc($today->new_delta("-1day")); 
say $today->printf("today: %Y-%m-%d %A\n"), $sunday->printf("Sunday: %
+Y-%m-%d %A");"
[download]

The complicated date formula calculates the Sunday of this week. It's complicated because you seem to want to work with weeks starting with Sundays, whereas I only know how to ask Date::Manip to work with week starting with Mondays.