Adam Dell and Padma Lakshmi in Feb. 2009
03/15/2012 AT 04:15 PM EDT
Padma Lakshmi's longstanding battle with her daughter's father has finally been resolved.
The Top Chef
host and Adam Dell, father of 2-year-old Krishna, have been embroiled in familial drama since the birth of their baby girl.
Venture capitalist Dell, 42, sued Lakshmi
, 41, last year for custody of their daughter Krishna, claiming that the reality TV star aimed "to minimize, if not eliminate, [Dell's] role in Krishna's life." (Lakshmi, who was in a relationship with the late businessman Teddy Forstmann,
would not publicly acknowledge who fathered her baby. Dell's lawsuit cited a DNA test that proved he was the father.)
According to Thursday's New York Post
, the pair have settled their dispute – and their child's name reflects the agreement.
Their daughter's full name will now be Krishna Thea Lakshmi-Dell. "Padma basically said it was fine to add his name, but she retains all decision-making power, which is really what matters most to her," a source close to Lakshmi tells PEOPLE. "She makes all the decisions on education, health, everything."
Dell and Lakshmi, who settled out of court, also came to a compromise on custody and visitation rights.
"Having her daughter spend time with her father was never a big issue for her," the source says of Lakshmi. In terms of visitation, "she gave a little more time because as the child gets older, it's normal to give more."
"It's been very difficult for her," adds the source. "She's happy that its over, that it was settled out of court and that nobody won or lost."
When reached, Dell's lawyer refused to comment "due to a confidential settlement agreement." Lakshmi's lawyer had no comment.