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Cooperative control of Connected and Autonomous Vehicles (CAVs) promises great benefits for mixed traffic. Most existing research focuses on model-based control strategies, assuming that car-following dynamics of human-driven vehicles are explicitly known. In this paper, instead of relying on a parametric car-following model, we introduce a data-driven predictive control strategy to achieve safe and optimal control for CAVs in mixed traffic. We first present a linearized dynamical model for mixed traffic systems, and investigate its controllability and observability. Based on these control-theoretic properties, we then propose a novel DeeP-LCC (Data-EnablEd Predictive Leading Cruise Control) strategy for CAVs based on measurable driving data to smooth mixed traffic. Our method is implemented in a receding horizon manner, in which input/output constraints are incorporated to achieve collision-free guarantees …