The design of an autopilot for a space launcher during the atmospheric stage is known to be a hard control problem, which contains an important number of conflicting specifications. This difficulty is increased by the presence of flexible modes, some of them acting inside the bandwidth, and parameter uncertainties. To solve such a problem, a multi-objective approach is proposed in this paper, which combines the Youla parameterization, suitable translations of the different specifications into linear matrix inequalities (LMI), and convex optimisation using a cutting plane algorithm (CPA). These tools together can handle high order controllers without numerical difficulties. They can also efficiently provide an answer to the crucial question of the feasibility or unfeasibility of the specifications. Although they can be used for a wide class of multi-objective problems, the paper doesn't particularly focus on the theoretical developments (which can be found in recently published works) but rather on their application to the space launcher control problem.