According to canon, the wand's allegiance is passed down to the person who defeats the original owner. We have many examples of that (An assassin murdering one of the Peverell Brothers for the Elder Wand; Harry disarming Draco for both the Elder Wand and Draco's Wand). The allegiance only passes down if it's taken without the owner's consent (Dumbledore planned on having the wand's allegiance die with him, as his Death by Snape was consented).
But what would happen if, let's say, Voldemort had used the Imperius curse on Draco, and Draco disarmed Dumbledore?
On one hand, Draco would have been the one disarming Dumbledore, but then again, it wasn't of his own volition, as Voldemort would have been the one controlling Draco. In that case, who would be owning 1/3 of the Deathly Hallows?
Preferably canon answers