Gemara

(Mishnah): One can force his slave (to transgress Nezirus), but he cannot force his wife.

Nedarim 77b (Beraisa): One may not say to his wife on Shabbos '(your vow is) annulled for you' or 'void for you' like he says during the week. Rather, he tells her 'take and eat (what she forbade)' or 'go drink', and the vow is void.

(R. Yochanan): He must be Mevatel the vow in his heart.

(Beraisa - Beis Shamai): On Shabbos, a man is Mevatel his wife's vow in his heart. During the week, he must say it;

Beis Hillel say, both of these are the same. It suffices to be Mevatel the vow in his heart. He need not say it with his lips.

Rishonim

The Rif (Nedarim 24b) brings the Gemara in Nedarim verbatim.

Ramban: One can force his wife only on Shabbos.

Rambam (Hilchos Nedarim 13:4): One who is Mevatel the vows of his wife or daughter need not say anything, and all the vows are Batel.

Rambam (5): The Bitul is to force her to do something she forbade. However, when he annuls he does not force her; he leaves her to do as she wants.

Rambam (6): If he gave to her food or drink and said 'take and eat or drink', she eats or drinks, and the vow is Batel automatically.

Rambam (7): One who is Mevatel need not say the words with his lips, rather, he is Mevatel in his heart and forces her to do (unlike her vow). Whether or not she did, the vow is Batel.

Rebuttal (Ra'avad): A Mishnah (Nazir 62b) teaches that one can force his slave (to transgress Nezirus), but he cannot force his wife. Rather, on Shabbos when he cannot explicitly annul, he tells her to eat in lieu of Hafarah, and annuls in his heart.

Ritva (DH Tanya and DH Chomer): The Mishnah teaches that it suffices to force a slave, but one can force his wife only if he also is Mevatel in his heart.

Defense (Kesef Mishneh): The Gemara did not say that Bitul works only on Shabbos. Rather, on Shabbos he must annul in a different way!

Radvaz: The Beraisa forbids saying to one's wife on Shabbos like he says during the week. Why must he say anything? All agree that on Shabbos, he can Mevatel in his heart! The Ra'avad explains that Beis Shamai and Beis Hillel refer to this Beraisa, and argue about what is needed in addition to telling her 'take and eat.' The Rambam must explain that 'take and eat' helps even during the week, and the Beraisa requires saying this differently on Shabbos than during the week. The Ra'avad understood that the Rambam requires physically forcing her to transgress. This cannot be, for he says 'whether or not she did (unlike her vow), it is Batel'! Rather, 'forcing' means telling her to eat.

Perush ha'Mishnayos: One does not say to his wife on Shabbos 'Mufar Lechi' like he says during the week. Rather, he tells her 'take and eat', and the vow is Batel automatically. He must Mevatel in his heart. If he cannot Mevatel by forcing her through action, he is Mevatel in his heart. The Tosefta (in our texts, a Beraisa (77b)) says that this suffices. The Gemara said that Hafarah in his heart does not help. This is because Hafarah returns the matter to the initial state; she can do as she wishes. Bitul is harder than Hafarah, for he must force her to do what she swore not to do; he need not verbalize it.

Merumei Sadeh (Nedarim 77b): The Rambam holds that he cannot force her to eat. He forces her not to conduct according to her vow.

Lechem Mishneh: The Rambam came to answer the Ran's question, that Hafarah in his heart does not help. Telling her to eat shows that he does not want the vow; it is in place of explicit Hafarah. Perush ha'Mishnayos connotes that he must Mevatel in his heart only if she did not eat. The Ra'avad seems to be unlike Beis Hillel, who do not distinguish Shabbos from weekdays! Perhaps the Ra'avad holds like the Re'em, who says that he says the Bitul inaudibly with his lips. Sometimes the Gemara calls this 'in his heart.'

Emek ha'Netziv (On Sifri, 2 p.286): The Rambam holds that when he intends to force her, mere intent helps. He must be Mevatel in his heart if he is unable to force her at the time.

Rosh (Nedarim 10:10): The Mishnah in Nazir teaches that one cannot force his wife. If she vowed not to eat neat or drink wine and he told her to eat and drink, this is not Hafarah. It helps only if he annuls in his heart, like the Re'em says.

Chasam Sofer (Nedarim 76b DH Mefirin): The Rambam holds that Hafarah in the heart does not help, but forcing with intent to annul works. He learns from "v'Im Hafer Yafer." The Ran holds that Hafarah in the heart works together with forcing. The Re'em holds that he commands her only to inform that he annulled. If thought alone helped, on Shabbos he would not be allowed to command her to eat! Rather, he requires a soft Dibur, like the Dibur for Pigul. (Perush ha'Rosh 77b DH v'Omar - he says in his heart, but not with his lips in a loud voice, 'it is annulled for you.' It need not be audible to his ears.)

Poskim

Shulchan Aruch (39): As long as one did not annul his wife's vow, he cannot force her to transgress, unless he told her 'take and eat' and thought in his heart the words of Hafarah. The Rambam holds that one who is Mevatel the vows of his wife or daughter need not say anything, and all the vows are Batel. The Bitul is to force her to do something she forbade. However, when he annuls he does not force her; he leaves her to do as she wants. E.g. if she vowed or swore not to eat or not to drink, and he said 'Mufar Lechi', it is annulled, and she may eat or drink. If he gave to her food or drink and say 'take and eat or drink', she eats or drinks, and the vow is Batel automatically.

Shulchan Aruch (40): One who annuls the vows of his wife or daughter must say the words with his lips. If he annulled in his heart, it is not annulled. One who is Mevatel need not say the words with his lips, rather, he is Mevatel in his heart and forces her.

Rema: This is like the Rambam. The first opinion (in Sa'if 39) does not distinguish Hafarah from Bitul. In either case it is not annulled or Batel until he (verbally) annuls, or is Mevatel in his heart and tells her 'take and eat.' This opinion is primary.