Among the renewable energy technologies, offshore wind energy is expected to provide a significant contribution for the achievement of the European Renewable Energy (RE) targets for the next future. In this framework, the increase of generated power combined with the alleviation of vibratory loads achieved by application of suitable advanced control systems can lead to a beneficial LCOE (Levelized Cost Of Energy) reduction. This paper defines a control strategy for increasing floating offshore wind turbine lifetime through the reduction of vibratory blade and hub loads. To this purpose a Proportional-Integral (PI) controller based on measured blade-root bending moment feedback provides the blade cyclic pitch to be actuated. The proportional and integral gain matrices are determined by an optimization procedure whose objective is the alleviation of the vibratory loads due to a wind distributed linearly on the rotor disc. This control synthesis process relies on a linear, state-space, reduced-order model of the floating offshore wind turbine derived from aero-hydroelastic simulations provided by the open-source tool OpenFAST. In addition to the validation of the proposed controller, the numerical investigation based on OpenFAST predictions examines also the corresponding control effort, influence on platform dynamics and expected blade lifetime extension. The outcomes show that, as a by-product of the alleviation of the vibratory out-of-plane bending moment at the blade root, significant reductions of both cumulative blade lifetime damage and sway and roll platform motion are achieved, as well. The maximum required control power is less than 1% of the generated power.