Authorities said he was wanted in connection with multiple armed sexual assaults including a burglary and sexual assault in Los Angeles last fall.
An FBI spokeswoman did not respond to messages asking whether the man exchanged gunfire with agents.
Carlson, who the agency said had been linked by DNA to sexual assaults dating back to 2003, had previously been tracked to South Carolina, Alabama and Florida, the FBI said.
Authorities say Carlson committed a burglary on July 13, 2017, during which he attempted to sexually assault a woman while using a weapon.
In September, he was charged with assault with intent to commit rape, assault with a deadly weapon and burglary, and he was arrested by Los Angeles police.
Carlson posted bond and was released, then fled to Mount Pleasant, South Carolina.
The FBI said investigators believe he then left the state in a car with a stolen gun and a significant amount of cash.
He was spotted in Hoover, Alabama, on Nov. 22 and led police on a high-speed chase, but officers called off the pursuit because of danger to the public.
Carlson was then spotted in Florida, first in Jacksonville on Nov. 28 and in Daytona Beach two days after that.
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(With inputs from APTN)