Connected and automated vehicles (CAVs) have a great potential to actively influence traffic systems. This has been demonstrated by large-scale numerical simulations and small-scale real experiments, whereas a comprehensive theoretical analysis is still lacking. In this paper, we focus on mixed traffic systems with one single CAV and heterogeneous human-driven vehicles, and present rigorous controllability analysis and optimal controller synthesis. Using the PBH controllability criterion, we reveal controllability properties of a linearized mixed traffic system in a ring road. It is proved that the mixed traffic flow can be stabilized by one single CAV under a very mild condition. We formulate the problem of designing CAV control strategies under a pre-specified communication topology as structured optimal controller synthesis. This formulation considers a system-level performance index that allows the CAV to actively dampen undesired perturbations in traffic flow. Numerical experiments verify the effectiveness of our results.