Mich. Comp. Laws § 722.531
Whenever it appears to the court that a child placed with, committed to, or released to a person, society, organization, association, or corporation licensed and approved by the family independence agency to receive, maintain, or place out for adoption minor children or to obtain homes for dependent and neglected children, is, by reason of mental or physical disability or for any other reason, incapable of adoption, the child may be committed to a proper state institution. A commitment under this section may be made by the family division of the circuit court for the county in which the home of the person, or the principal office of the society, organization, association, or corporation is located, or for the county in which the child was placed with, released to, or committed to the person, society, organization, association, or corporation, or for the county in which the child may be found, upon application of the person or the principal officer of the society, organization, association, or corporation to which such child has been committed or released. The expense of committing the child to a state institution and any expense chargeable to counties for maintenance in a state institution shall be paid by the county where the child was a resident at the time of his or her placement with, release to, or commitment to the person, society, organization, association, or corporation.
MCL 722.531