Modelling and boundary control of extensible and shearable slender curved beams in flow with large in-plane deflection are addressed in this paper. Equations of motion of the beams are derived based on the difference between purely geometric deformation and actual deformation from the reference and actual configurations to a virtual straight configuration. Two control designs based on the Lyapunov direct method are proposed for curved beams with both small and large curvature. The proposed boundary controllers guarantee globally (practically) kappa(infinity)-exponential stabilisation of the beam's motions at the origin. Two Lyapunov-type theorems are developed for nonlinear evolution systems in Hilbert space for analysis of well posedness and stability of the closed-loop system.