## 1 Introduction and Motivation  
According to statistics of the DGT (_Spanish National Traffic Administration_), around 91% of fatal traffic accidents take place on highways and conventional roads [1] in Spain. Up to 71% of these accidents occur due to the collision of two or more vehicles in transit, against a suddenly appearingobstacle on the road, or because of pedestrians or animals running over. This information shows how relevant it is to develop new technologies that can assist the driver or make autonomous decisions to increase passengers' safety. In this regard, the automotive industry is currently moving towards the automation of every single aspect of the driving experience [2]. The main trend, before cars are fully automated, consists of improving the driver assistance by providing new mechanisms that can increase safety and comfort. This can be effectively implemented by the installation of devices that can monitor the environment continuously and, at the same time, determine under which situations the driver needs some kind of specific aid (through somewhat advanced smart processing applications). By monitoring the environment, we mean using specific cameras and/or sensors that can acquire useful data to improve the driving experience. Communications will also play an important role when disseminating information monitored by sensors installed on board to enhance the safety conditions of passengers while driving. The standard WAVE 1609/802.11p [3] has been designed for this purpose, and it covers the general communication schemes that vehicles will implement in to support V2V (_vehicle to vehicle_) and V2I (_vehicle to infrastructure_) communications. Through the use of communications, the amount of information that a car may process is much higher, and thus, with a proper processing scheme, it can be really useful to make timely and smarter decisions that would not be possible when isolated. Therefore, by using such technologies, vehicles could benefit from the environmentally collected data to take autonomous actions in order to reduce the probability of a fatal event (vehicle-vehicle collision, pedestrian/animal running over) at high speed. Within this topic, a very concerning problem is how to perform the best emergency maneuvers. Interest in this issue has increased in the last decades [4], resulting in methods to find optimal trajectories for different objectives, mainly focused on single vehicles. However, emergency maneuvers should be planned by taking into account surrounding vehicles for obvious reasons. The deployment of vehicular communications paves the way for the development of methods to compute optimal trajectories for a set of involved vehicles. This is certainly a challenging goal. First, there are a wide number of possibilities to state the problem and select the appropriate functional to optimize. Second, either the problem may be solved in a distributed way or a leader car in a clustering scheme may solve it and deliver the solution to the remaining involved vehicles. A hybrid method entailing the exchange of independent solutions may be also suitable. Third, the variability to which the shape of trajectories might be subject (due to the disturbances in the controls that vehicles might suffer) makes it necessary to use mechanisms to correct the possible deviations from the optimum trajectory while maneuvering to reduce the negative influence of undesirable phenomena. In the present study, we focus on the development of useful procedures to dynamically optimize the trajectory of a vehicle under timing constraints, _i.e._, changing its lateral position on the road to maximize the lateral distance within an interval of \(t_{f}\). Additionally, we minimize some other mobility parameters like the lateral speed at the end of the path (to reduce the impact of sudden inertial changes). Finally, we carry out a study on how diversion from the optimum trajectory due to Gaussian noise in measurement sensors can be minimized thanks to the well-known the Kalman Filter technique [5].