How does Prospero have control over Caliban?
Caliban himself explains how it is that Prospero came to have control over him. We know that Caliban is at least partly animalistic, perhaps more animal than human; he is described as having been "litter'd" like a puppy, the offspring of Sycorax the witch and the Devil.
Like an animal, then, he responds to affection and empathy—in act 1, scene 2, Caliban says (speaking to Prospero) that when Prospero first arrived, "thou strok'st me and made much of me," like a man trying to gain the favour of a cat. This worked—Caliban was then happy to show Prospero all the "fruits" of the island, in return for what seems to be Prospero teaching Caliban language: "how to name the bigger light and how the less/That burned by day and night."
After this, Caliban says, he loved Prospero; before Prospero, we can infer, nobody had ever showed Caliban kindness or attention, and Caliban responded in kind, learning readily what he was taught and teaching Prospero how to survive on the island.
Of course, this state of happiness does not last, as it seems Caliban is unable to control his animalistic nature, but, with Prospero at least, Caliban's animalism drives him to respond to fondness with fondness.
The relationship between Caliban and Prospero starts off on a reasonably positive note. Prospero shows Caliban affection, and Caliban himself speaks of feeling love for Prospero. So at first, Prospero is able to control Caliban the way anyone who is on the receiving end of an emotional attachment can control the one who loves. Prospero also goes on to try to educate Caliban and to teach him how to speak, so the power dynamic grows in Prospero's favor as he is the teacher to Caliban's pupil.
Everything changes when Caliban tries to rape Miranda, Prospero's daughter, and the tables turn for a time, when Caliban is revealed to have some power over Prospero through Caliban's potential strength and potential ability to overpower Miranda and hurt her. Then Prospero resorts to control by force and fear, punishing Caliban for his crime with intimidation, spells, and physical pain.
Caliban is the son of Sycorax, a witch who had controlled the island before Prospero got there. We are not exactly told how Prospero got control of the island, but since he landed there, he has been the one in control of the island.
Prospero is able to control the spirits of the island. By using these spirits, he is able to control Caliban as well. If you look at the parts where Caliban talks to Stephano, you can see him talking about how Prospero uses spirits to control him -- he talks about how they will hurt him if he does not obey Prospero.
Prospero has clearly got some control over Caliban. However the extent of control is substantially lower than you would have thought. In act 1 scene 2, Miranda and Prospero visit Caliban. Miranda obviously doesn’t care about control over him, “I do not love to look on,” all she wants is too stay away from him. This is not her only vain moment in the play. Prospero however makes a great show of how much power he has, how he can punish Caliban, yet Caliban is completely used to this form of abusive behaviour that he replies with a non-sequitur. He is obviously immune to his punishments by now. That shows that he just ignores Prospero most of the time, and doesn’t see him as a threat. However Prospero can control some aspects of Calibans life. He controls where Caliban lives, what Caliban does, and where he is allowed to go. Prospero cannot control what Caliban says to him. Prospero taught him their language, so Caliban must have heard it from Prospero first. In conclusion, Prospero controls Caliban in the only way he knows how, with magic.