The influence of inter-vehicular communication on platoon control performance is studied by modeling the communication channel as an analog fading channel. First, the conditions for achieving platoon mean square stability (MSS) are derived for three cases: undirected information flow (UIF) topology with identical fading channels, balanced directed information flow (BDIF) topology with identical fading channels, and general information flow topology with non-identical fading channels. Second, in order to mitigate the impact of fading channels on platoon MSS, some feasible controllers are proposed for each one of the three cases by solving the corresponding modified Riccati inequalities (MRIs). Finally, simulations are conducted to compare three different fading channels (Nakagami channel, Rician channel and Rayleigh channel), and the results verify the superiority and effectiveness of the provided control methods in ensuring the platoon MSS. Both theoretical analysis and simulation results demonstrate that the influence of fading channels on platoon MSS is determined by the statistics of fading channels, and the designed controllers can suppress the impact of fading channels on the stability of vehicle platoon.