Electrostatically actuated mesh reflectors are a promising technology for space-based sensing and communication antennas. Key advantages over current passive mesh reflectors include the ability to correct surface distortions (e.g., from manufacturing defects and environmental disturbances) and to achieve beam steering and beam shaping via dynamic adjustment of the reflector surface. Implementation of an electrostatically actuated mesh requires a method for determining optimal control voltages that achieve a desired reflector shape. This work demonstrates such an inverse method using a simplified numerical model of a linear isotropic elastic membrane subject to nonlinear electrostatic pressure. The model is used to determine the number of electrodes required to achieve a desired surface error. A prototype reflector, consisting of a 0.5-m diameter gold mesh and 9 electrodes, is constructed and various reflector shapes, including non-axisymmetric ones, are produced and refined using an open-loop controller based on the method. Before refinement, the experimentally measured RMS surface error is within 840μm of the target shape. The method is also shown to be useful for fine-tuning the shape; with only 3 iterations of surface refinement, the RMS error is reduced to within 231 μm. The experimentally characterized surface error proves that the method can adapt to unmodeled mechanics, achieving the desired shape precision for operation at 35 GHz Ka-band. Full-wave electromagnetic simulations of the target and experimental reflector surfaces were performed, and it was found that the fine-tuned surfaces had sufficient surface precision for convergence of the main lobe of the antenna pattern. This work demonstrates both the viability of, and method for achieving, shape control of an electrostatically actuated reflector.