Each child or youth in the custody of the Commissioner of Children and Families due to behavioral problems, and served by an approved residential treatment facility with special education programs, may continue in such facility beyond eighteen years of age until his or her program is completed. The Commissioner of Mental Health and Addiction Services shall be responsible for the payment of board and care costs for any child or youth who remains in an approved residential treatment facility with special education programs beyond eighteen years of age pursuant to this section provided such child or youth meets the eligibility requirements established by the commissioner.