The dependence of autonomous vehicles (AVs) on sensors and communication links exposes them to cyber-physical (CP) attacks by adversaries that seek to take control of the AVs by manipulating their data. In this paper, the state estimation process for monitoring AV dynamics, in presence of CP attacks, is analyzed and a novel adversarial deep reinforcement learning (RL) algorithm is proposed to maximize the robustness of AV dynamics control to CP attacks. The attacker's action and the AV's reaction to CP attacks are studied in a game-theoretic framework. In the formulated game, the attacker seeks to inject faulty data to AV sensor readings so as to manipulate the inter-vehicle optimal safe spacing and potentially increase the risk of AV accidents or reduce the vehicle flow on the roads. Meanwhile, the AV, acting as a defender, seeks to minimize the deviations of spacing so as to ensure robustness to the attacker's actions. Since the AV has no information about the attacker's action and due to the infinite possibilities for data value manipulations, each player uses long short term memory (LSTM) blocks to learn the expected spacing deviation resulting from its own action and feeds this deviation to a reinforcement learning (RL) algorithm. Then, the attacker's RL algorithm chooses the action which maximizes the spacing deviation, while the AV's RL algorithm seeks to find the optimal action that minimizes such deviation. Simulation results show that the proposed adversarial deep RL algorithm can improve the robustness of the AV dynamics control as it minimizes the intra-AV spacing deviation.