Core can certainly do this, but it’s quite complicated to set up and it might be overkill for the particular scenario.
Instead, you can consider setting up a virtual timer. There is an article in the community – created wiki for how to do this.
Once you have the virtual timer, you have the presence sensor arrival after 1130 turn the virtual timer on for 15 minutes and then turn itself off. And you will have a Smart lighting automation so that the real light follows the timer, turning itself on when the timer comes on and turning itself off when the timer goes off.
That way your original time schedule rule still works, but you also have a virtual timer with a different rule in a different time period. 
http://thingsthataresmart.wiki/index.php?title=How_to_create_a_virtual_timer_for_a_light
And if you do decide you want to look into core, here’s the FAQ for starting that. But it does sound like all you need is a virtual timer.