The phrase "keep under" means to control or restrain someone or something. Synonyms for this phrase include manage, restrain, control, dominate, regulate, suppress, subdue, and contain. When you want to manage or control a situation, you can use these synonyms to express yourself in a more diverse way. For example, you can say "I need to regulate my spending" instead of "I need to keep my spending under control." Similarly, instead of saying "the police kept the protesters under control," you can use "the police suppressed the protesters" or "the police managed the protesters." These synonyms can help you to express yourself more precisely.