Bistable structures are a type of morphing structure which have the ability to obtain two stable shapes which do not require any energy to be maintained. In this work, a reference governor control scheme is implemented to control the shape change of a bistable structure while satisfying actuator saturation constraints. The structure is modelled as the nonlinear Duffing-Holmes oscillator and the actuation force becomes a polynomial of the states once the system is linearized. By extending an augmented version of the state, the computation of the maximal output admissible invariant set in the case of polynomial actuator constraints is achieved. The reference governor is implemented by solving a nonlinear optimization problem at initialization and then uses a bisection algorithm at subsequent time steps. This methodology is shown to be an effective means by which a bistable structure may be controlled despite the polynomial constraints associated with actuator saturation. The proposed methodology is then compared to the most current method for controlling bistable structure shape change.