O.C.G.A. § 14-11-1102 (2019)

(a) If each person required by Code Section 14-11-205 to execute any document fails or refuses to do so, any other person who is adversely affected by the failure or refusal may petition the superior court of the county where the registered office of the limited liability company is located to direct the execution of the document. If the court finds that it is proper for the document to be executed and that every person so designated has failed or refused to execute the document, it shall order the Secretary of State to file the document in appropriate form notwithstanding the lack of required execution.

(b) The court shall assess the costs and expenses of such proceeding against the limited liability company, except that all or any part of such costs and expenses may be apportioned and assessed, as the court may determine, against any or all of the persons required by Code Section 14-2-205 to execute a document who failed or refused to do so if the court finds that such failure or refusal was arbitrary, vexatious, or otherwise not in good faith.