Disciplining children on Shabbos:

It is permitted to discipline or punish children on Shabbos.[1] One may thus lock a child in his room for disciplinary purposes.[2] Nonetheless, it is possibly even Biblically forbidden to hit or spank a child if doing so can cause a bruise to the child.[3] Furthermore, it is not in the spirit of Shabbos to show anger or fury, or give punishment even for the sake of Chinuch, as learned from the reason why the Torah prohibited a Beis Din from giving punishment on Shabbos. The Torah desired that Shabbos be a day of rest for all, even those who deserve punishment, and hence one is to abstain from punishing or showing any anger on Shabbos to the utmost extreme.[4]

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[1] Setimas Hapoksim as the above prohibition is only in reference to a Beis Din, or community, punishing or incarcerating an individual, and not to a parent who does so for educational purposes.

[2] Tzitz Eliezer 15:41 brought in Piskeiy Teshuvos 316:23

The trapping prohibition: There is no trapping prohibition involved in trapping humans. [Setimas Hapoksim in 339; Piskeiy Teshuvos 316:23] One may thus lock a child in his room if needed. [Tzitz Eliezer 15:41 brought in Piskeiy Teshuvos 316:23]

Other opinions: Some Poskim question that perhaps the trapping prohibition applies to a child. [Avnei Nezer 189:22 based on Tosafus Menachos 64a]

[3] Some Poskim suggest that perhaps giving lashes is Biblically forbidden because it may cause a wound, which is a Biblical prohibition, and is thus included in the Biblical prohibition against giving punishment. [Rambam 24:7 as explained in M”A ibid; M”A 278:1 in name of Turei Even 40 and Rashal regarding a fight that this applies even according to Rebbe Shimon who holds Eino Tzarich Legufa is exempt; P”M 339 A”A 3] However, in truth, perhaps it is only a Rabbinical prohibition, as it is a Melacha She’eiyno Tzarich Legufa and one has no intent to make a wound. [see Machatzis Hashekel on M”A ibid that he has no intent to make a wound; P”M ibid that the above Biblical prohibition only applies according to opinion of Eino Tzarich Legufa Chayav; Admur 316:16]

[4] Piskeiy Teshuvos 339:7