In this paper, the H∞ bumpless transfer reliable control problem of Markovian switching linear parameter-varying (LPV) systems subject to actuator failures is addressed. A bumpless transfer reliable control constraint is introduced to describe the transient performance for such systems. By the use of the parameter-dependent Lyapunov function approach to reduce control bumps generated by the Markovian switching, a reliable switching controller which is also parameter-dependent is constructed to attenuate external disturbances. A criterion checking the solvability of the H∞ bumpless transfer reliable control problem is provided. As an application, a turbofan-engine model is used to verify the validity of the proposed design method.