The emergence of autonomous vehicles is expected to revolutionize road transportation in the near future. Although large-scale numerical simulations and small-scale experiments have shown promising results, a comprehensive theoretical understanding to smooth traffic flow via autonomous vehicles is lacking. Here, from a control-theoretic perspective, we establish analytical results on the controllability, stabilizability, and reachability of a mixed traffic system consisting of human-driven vehicles and autonomous vehicles in a ring road. We show that the mixed traffic system is not completely controllable, but is stabilizable, indicating that autonomous vehicles can not only suppress unstable traffic waves but also guide the traffic flow to a higher speed. Accordingly, we establish the maximum traffic speed achievable via controlling autonomous vehicles. We also design an optimal control strategy for autonomous vehicles to actively dampen undesirable perturbations. These theoretic findings validate the high potential of autonomous vehicles to smooth traffic flow. * Electronic address: yang.zheng@eng.ox.ac.uk