Abstract: The attitude motion of the flexible spacecraft is strongly coupling with the vibration of the flexible components, which will result in complicated nonlinear system characteristics. Because of the nonlinearity of the flexible spacecraft, the dynamics modeling and the control of its attitude motion are challenging problems. This paper aims to present a data-driven modeling method based on the Koopman operator theory for the flexible spacecraft, and design the LQR optimal controller based on the model obtained by data-driven identification for the attitude motion and flexible vibration suppression. Firstly, based on Koopman operator theory and sparse identification for nonlinear dynamics algorithm (SINDY), a data-driven identification method for the attitude dynamics of flexible spacecraft on SO(3) is proposed. According to the dynamics characteristics of the flexible spacecraft attitude dynamics on SO(3), a set of observables containing the original states of attitude dynamics is designed to identify the generalized linear model of the flexible spacecraft attitude dynamics on lifting space spanned by those observables. Secondly, the global linearization is carried out based on the Koopman operator theory. Under the assumption of small angular velocity, by removing the high-order terms in the generalized linear model, the finite-dimensional Koopman sparse model is obtained. The predict ability of such a linear model obtain by SINDY and Koopman linearized model is verified by simulation. Finally, on the basis of the Koopman linearization model identified from the data, an optimal linear quadratic regulator (LQR) controller based on Koopman operator is proposed for attitude motion control and vibration suppression of the flexible spacecraft. The effectiveness of the proposed controller is verified by simulation, and the proposed controller is compared with the classical nonlinear optimal control method to show its advantages.