The phrase "kept under" typically refers to something that is being controlled, monitored, or restricted. Synonyms for this phrase can include "supervised," "watched," "managed," "regulated," "restrained," "controlled," "contained," "governed," "held in check," "kept in restraint," "reined in," and "curbed." Each of these synonyms conveys a slightly different emphasis on the level of control or restriction being exerted, but they all suggest that something is being carefully managed or monitored. Whether referring to a person, animal, or object, the phrase "kept under" can be a useful shorthand for describing a situation in which close oversight is required to ensure safety or order.