Rumor spreading on social media platforms can significantly impact public opinion and decision-making. In this paper, we proposed an innovative ignorant-spreader-expositor-hibernator-remover (ISEHR) rumor-spreading model with multivariate gatekeepers. Specifically, by analyzing the model's dynamics, we identified the critical threshold that determined the persistence or extinction of rumor spreading. Moreover, we applied the Routh-Hurwitz judgment, Lyapunov theory, and LaSalle's invariance principle to investigate the existence and stability of the rumor-free/rumor equilibrium points. Furthermore, we introduced the optimal control to alleviate rumor spreading with the multivariate gatekeeper mechanism. Finally, extensive numerical simulations validated our theoretical findings, providing insights into the complex dynamics of rumor spreading and the effectiveness of the proposed control measures. Our research contributes to a deeper understanding of rumor spreading on social networks, offering valuable implications for the development of effective strategies to combat rumor.