The petition, filed on Wednesday in the probate division of Los Angeles County Superior Court, asks that Wilson's publicist-manager, Jean Sievers, and his business manager, LeeAnn Hard, be appointed "co-conservators of his person."
The assets of the 81-year-old singer-songwriter's estate are already held in trust, according to the petition.
An accompanying declaration filed by a physician, Dr. Stephen Marmer, states that Wilson suffers from a "major neurocognitive disorder" and is taking a medication for treatment of dementia.
The doctor said Wilson lacks the capacity to give informed consent to any form of medical treatment.
The musician's wife, Melinda Ledbetter Wilson, who died in January at age 77, had been her husband's daily caregiver. Her death left him "unable to properly provide for his own personal needs for physical health, food, clothing or shelter," according to the petition.
The appointment of Sievers and Hard will ensure that Wilson's "daily living needs are satisfied and that he has the best possible care while remaining in his home," the petition states.
The decision to seek a conservatorship was made "after careful consideration and consultation among Brian, his seven children" and his doctors, Wilson's family said in a statement posted on his official website.
Sievers confirmed in a statement on Friday to Reuters, initially given to the New York Times, that Wilson had been diagnosed with dementia, and that she and Hard would ensure that his daily needs are met while "he continues to lead an active life."
"Brian is doing very well," she added.
A hearing on the petition is set for April 30, but Dr. Marmer requested Wilson be excused from attending the proceeding because going to court "would be emotionally very stressful, physically difficult and detrimental to his health."
Wilson created some of rock's most enduring songs, including Good Vibrations, God Only Knows, California Girls and Surfin' U.S.A., in a career dogged by decades-long bouts with substance abuse and mental disorders.
In June 2019, Wilson postponed a US summer tour, citing concerns about his mental health, saying he was "struggling with stuff in my head and saying things I don't mean, and I don't know why" in the aftermath of a series of back surgeries.
He formed the Beach Boys in the early 1960s with younger brothers Carl and Dennis Wilson, cousin Mike Love and friend Al Jardine in their hometown, the Los Angeles suburb of Hawthorne.
The band went on to score three dozen top-40 hits celebrating Southern California's sunny youth culture, with Wilson writing and composing most of the early works.
Their landmark 1966 album Pet Sounds was considered Wilson's magnum opus and was ranked by Rolling Stone magazine in 2012 as second only to the Beatles' Sgt. Pepper's Lonely Hearts Club Band on its list of the 500 greatest rock albums.