"Control" in general is not particularly well defined as a yes/no proposition. You can likely rigorously define an agent's control of a resource by finding the expected states of that resource, given various decisions made by the agent.
That kind of definition works for measuring how much control you have over your own body - given that you decide to raise your hand, how likely are you to raise your hand, compared to deciding not to raise your hand. Invalids and inmates have much less control of their body, which is pretty much what you'd expect out of a reasonable definition of control over resources.
This is still a very hand-wavy definition, but I hope it helps.
Any scenario where advanced AI takes over the world requires some mechanism for an AI to leverage its position as ethereal resident of a computer somewhere into command over a lot of physical resources.
One classic story of how this could happen, from Eliezer:
You can do a lot of reasoning about AI takeover without any particular picture of how the world gets taken over. Nonetheless it would be nice to have an understanding of these possible routes. For preparation purposes, and also because a concrete, plausible pictures of doom are probably more motivating grounds for concern than abstract arguments.
So MIRI is interested in making a better list of possible concrete routes to AI taking over the world. And for this, we ask your assistance.
What are some other concrete AI takeover mechanisms? If an AI did not have a solution to the protein folding problem, and a DNA synthesis lab to write off to, what else might it do?
We would like suggestions that take an AI from being on an internet-connected computer to controlling substantial physical resources, or having substantial manufacturing ability.
We would especially like suggestions which are plausible given technology that normal scientists would expect in the next 15 years. So limited involvement of advanced nanotechnology and quantum computers would be appreciated.
We welcome partial suggestions, e.g. 'you can take control of a self-driving car from the internet - probably that could be useful in some schemes'.
Thank you!