What were Dumbledore's actual intentions for the Elder Wand?
Albus Dumbledore knew that if the elder wand goes into the wrong hands, the outcome will be really bad. So he came up with his brilliant plan. According to the legend, the ownership of the wand passes on to the person who defeats/ disarms the current owner in a duel. Dumbledore asked Snape to kill him in the astronomy tower when he was surrounded by the death eaters and Draco. Since here Dumbledore himself is choosing to die and is not defeated by Snape, the wand's ownership will remain with Dumbledore. This way he wanted to end the chain of ownership of the wand. That was his actual intention.
But obviously his plan went wrong when Draco disarmed him and the ownership passed to Draco. In the fight that took place in Malfoy mansion, Harry disarmed Draco. So finally the ownership passed on to Harry.