This paper addresses the fault-tolerant control issue for a class of flexible air-breathing hypersonic vehicles. Firstly, a longitudinal dynamic model with process faults is established, which contains an ordinary differential equation (ODE) for rigid body, an Euler-Bernoulli beam equation for flexible modes, and a new boundary connection between them; Secondly, a novel fault-tolerant control scheme is proposed to accommodate process faults and suppress vibrations, which relies on the direct Lyapunov method and the bilinear matrix inequalities (BMIs) technique; Thirdly, in order to compute the gain matrices of the fault-tolerant control law, a two-step algorithm is provided to solve the BMI feasibility problem in terms of linear matrix inequality optimisation technique. Finally, the simulation results are provided to illustrate the effectiveness of the theoretical results.