1. I let the system [to] perform the task.
2. I apply the method to the platform.
I would take #2 as a VP3, with "to the platform" as an adverbial prepositional phrase.
For #1, however, I might take "perform the task" as an object complement; but I'm not sure.
(Maybe another member will be prepared to offer a 2nd opinion.)
All the best,