He stipulated that his son be raised in New York, Chicago or San Francisco, or visit at least twice a year to be exposed to the "culture, arts and architecture" on offer.
Cooper, 10, is to inherit half his trust fund when he turns 25 and the rest when he is 30 years old, US media said.
O'Donnell lives with Cooper, and daughters Tallulah, 7 and Willa, 5, in Manhattan's West Village neighborhood.
The Oscar-winning actor was found dead in his rented apartment on February 2.
O'Donnell had raised the alarm after Hoffman failed to turn up as planned to see the children. The couple had recently split up, reportedly because of his drug use.
The 46-year-old Hoffman was hailed by many as the finest character actor of his generation.
He won an Oscar in 2006 for his portrayal of author Truman Capote in "Capote" and earned three further Academy Award nominations as best supporting actor in 2008, 2009 and 2013.
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Cooper, 10, is to inherit half his trust fund when he turns 25 and the rest when he is 30 years old, US media said.
O'Donnell lives with Cooper, and daughters Tallulah, 7 and Willa, 5, in Manhattan's West Village neighborhood.
The Oscar-winning actor was found dead in his rented apartment on February 2.
O'Donnell had raised the alarm after Hoffman failed to turn up as planned to see the children. The couple had recently split up, reportedly because of his drug use.
The 46-year-old Hoffman was hailed by many as the finest character actor of his generation.
He won an Oscar in 2006 for his portrayal of author Truman Capote in "Capote" and earned three further Academy Award nominations as best supporting actor in 2008, 2009 and 2013.
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