'exerts authority' is correct and usable in written English
You can use it when you want to describe someone or something having an influence over someone or something else. For example: "The headmaster exerts authority over his students."
Women of Ms. Huston's generation in Hollywood do not take opportunities to exert authority for granted.
The governor of the Bank of England had almost papal powers and, it was said, could exert authority by arching an eyebrow.
That could split the country or provoke intervention by the army.The government has neither exerted authority nor achieved compromise.
Guardiola has not been slow to show decisiveness and exert authority, banishing Joe Hart to Italy and leaving Yaya Touré out of his Champions League squad.
Maria Kowroski performed the ballerina role tenderly and lightly; she's reluctant to exert authority or find a wide range of color in the music.
Bishops and nobles often exerted authority over wealthy abbeys; the abbeys thus petitioned Sylvester to free them from all exterior control except that of the papacy.
But Mr. Tsvangirai has made the same demands privately to Mr. Mugabe, and so far the shared control of the Home Ministry and the police force it oversees has evidently left Mr. Tsvangirai unable to exert authority.
The Yemeni government has long struggled to exert authority in remote and mountainous areas like Abyan, a known refuge for militants and one of several provinces where Al Qaeda is believed to operate with relative impunity.
What results is a smoldering disrespect for the police and a challenge for the government of the president-elect, Vicente Fox Quesada, as he tries to exert authority and bolster the rule of law.
"Her Majesty's government takes a grave view of any attempt by Spain to exert authority or control within British Gibraltar territorial waters and considers such incursions as a violation of our sovereignty.
