We present two mathematical models of the DarkO tail-sitter convertible UAV developed and 3D printed at the École Nationale de l'Aviation Civile (ENAC), in Toulouse (France). During a hover flight, the UAV is vertical, which offers a large wing area facing the wind. Thus, aerodynamic disturbances have a strong influence on the nonlinear UAV dynamics. Our models capture this behavior and allow us to characterize relevant equilibria, in the presence of a constant wind, and the corresponding wind-dependent linearized dynamics. Using a parametric family of models, we design an optimality-based robust static output feedback controller that uses two degrees of freedom on the orientation to stabilize the UAV in a hovering condition in spite of an unknown constant wind. This control law has been implemented in the Paparazzi autopilot software to obtain experimental results that validate our theory.