Wiktionary
vb. (context idiomatic English) to pay careful attention to a situation or a thing, so that you can deal with any changes or problems.
Usage examples of "keep a close watch on".
Scott continues to keep a close watch on the matter-antimatter intermix for our warp engines—.
This, I'm sorry to say, is the case with the Baudelaire orphans and their plan to keep a close watch on Captain Sham, for after hours and hours of worrying they heard an enormous crash of shattering glass, and knew at once that keeping watch hadn't been a good enough plan.
It is to keep a close watch on the planet and to report at once if rocket experiments are in progress.
And therefore Riktors Ashen will be instructed to keep a close watch on you.
She might disdain the shallow gaiety of the festivities, but he should have known she'd want to keep a close watch on her cousin and his bride.