## 1 Introduction and Motivation
The goal of the recent research activities was to find a control design method with which many autonomous vehicles can be handled, while the energy consumption and time minimization requirements, along with safety, i.e., the avoidance of collisions, are ensured. Similarly, the purpose of this paper is to provide a novel solution to the recent problems through the following concept. The increase in the number of vehicles is handled through separation, i.e., the control designed for individual autonomous vehicles, which considers limited information on the environment. In the solution the performance requirements are priorized and the specifications are guaranteed through different methods, such as optimal control and RL-based techniques. Thus, in this way the advantages of each method, such as providing guarantees and optimization of the performance level, can be achieved. This paper focuses on the optimal design of the longitudinal control of an autonomous vehicle. Its elements and the related problems regarding the paper can be summarized as follows. * The objective may contain multiple criteria, the most important of which are the minimization of traveling time, energy consumption and the maximization of comfort. The variables of the optimization task are the control inputs of the vehicles, e.g., longitudinal acceleration commands or traction/braking forces. * The dynamics of each vehicle provide constraints in the optimization problem. * Furthermore, the safe motion of the vehicles, i.e., collision avoidance, must be guaranteed, which leads to constraints for vehicles with intersecting routes. * The velocity of the vehicles must be kept in a predefined range, considering the speed limit. * The control input of the vehicles must also be limited due to the physical limits of the driveline, the braking system and the tire-road contact. In this paper the optimization problem of the vehicle's motion is divided according to the performance aspects. First, a quadratic optimization problem is proposed, whose role is to guarantee collision avoidance and the limitation of the velocities of vehicles. The constraint of collision avoidance is formed through a linear approximation of the quadratic constraint, which leads to high efficiency in computation. The quadratic optimization is solved at every time step during the motion of the vehicles. Second, the advantages of reinforcement learning in the improvement of the economy performances, e.g., the minimization of the control input, are exploited. In the training process the previously formed quadratic optimization task is applied as a part of the environment for learning. Similarly, during the operation of the control system, the trained neural network and the quadratic optimization task operate together. The effectiveness of the resulting control system through simulation examples with various vehicles is analyzed. Thus, the contributions of the paper are as follows. The control design problem for autonomous vehicles in intersections is formed in a novel way through the division of the problem. It leads to the reduced complexity of the control problem in real-time computation. Moreover, an environment model for reinforcement learning is created, which guarantees that collision avoidance can be provided. An important benefit of the method is that the safety performance level of the resulting control structure is guaranteed; i.e., a collision of vehicles cannot be caused by the learning-based control agent. From an application point of view, an advantage of the method is that it is able to handle various situations at the intersection; i.e., the method is independent of the number of vehicles. Although this paper focuses on autonomous vehicle control, the proposed control design method may be used for further applications, considering the similarities with the intersections. For example, in the field of microfluid research, an actual problem is to find modeling and control methods for the handling of T-junctions [45].