This paper discusses a framework to design elements of the plant and control systems for floating offshore wind turbines (FOWTs) in an integrated manner using linear parameter-varying models. Multiple linearized models derived from high-fidelity software are used to model the system in different operating regions characterized by the incoming wind speed. The combined model is then used to generate open-loop optimal control trajectories as part of a nested control co-design strategy that explores the system’s stability and power production in the context of crucial plant and control design decisions. A cost model is developed for the FOWT system, and the effect of plant decisions and subsequent power and stability response of the FOWT is quantified in terms of the levelized cost of energy (LCOE) for that system. The results show that the stability constraints and the plant design decisions affect the turbine’s power and, subsequently, LCOE of the system. The results indicate that a lighter plant in terms of mass can produce the same power for a lower LCOE while still satisfying the constraints.