We model and analyze the influence of small amplitude transverse wall oscillations on the evolution of velocity perturbations in channel flows. The amplitude and frequency of periodic oscillations enter as coefficients, and spatially distributed and temporally varying body force fields enter as stochastic external excitations to our models. We quantify the effect of these excitations on velocity perturbation energy and develop a system theoretic paradigm for the optimal selection of transverse oscillation parameters for turbulence suppression. We use a perturbation analysis to demonstrate that depending on the wall oscillation frequency the energy of velocity perturbations can be increased or decreased compared to the uncontrolled flow. Our results provide a first compelling theoretical explanation as to why properly designed transverse wall oscillations can suppress turbulence in the wall-bounded shear flows.