## 4 Conclusions  
In this paper, we have discussed how to trace the optimum trajectory of a high speed vehicle that wants to relocate its lateral position before a time interval of \(t_{f}\). We have proposed and evaluated four different functionals and discussed their advantages and drawbacks. Functionals \(J_{D1}\) and \(J_{D2}\) provide a better stability during the trajectory because they only update the mobility parameters according to the final lateral position. \(J_{D3}\) and \(J_{D4}\), however, are based on updating the lateral position in terms of the instantaneous distance, which requires faster changes in the mobility evolution. On the other hand, whereas \(J_{D1}\) is really useful for higher values of \(t_{f}\) (since apart from reaching the optimum position it arrives with null lateral speed), \(J_{D2}\) reaches the last position with a speed always higher than zero. However, for low distances until the optimum position (i.e. low values of \(t_{f}\)), \(J_{D2}\) can get to this position earlier, while the other functional focuses its attention on reaching the last position at zero speed, not caring as much about how far is the vehicle from the optimum position. From these results, we can deduce that in general optimizing in terms of the final lateral distance can be better to avoid very rapid changes in acceleration, which could imply the need to have higher values for the discretization factor \(N\) (at the expense of more computation overhead). However, we have not explored the great multiplicity of scenarios that appears from this premise, therefore using functionals \(J_{D3}\) and \(J_{D4}\) (or a derivation of them) might be more convenient. In a future work, we will investigate a wider range of scenarios to completely characterize the performance of the four functionals (and possible extensions).  
On the other hand, the problem solved here corresponds to a scenario with no obstacles or other vehicles. It is our aim to study the problem for a higher number of vehicles (more differential equations) and obstacles. This results in functionals whose gradients are semi-convex. For this reason, depending on the initial configuration values, the _Gradient Projection Algorithm_ may not converge to the globalminimum. Formulating the functionals slightly differently or even using other mathematical techniques for solving this type of optimum control problems will probably be necessary.  
Last but not least, we have also carried out a first-step evaluation of how additive Gaussian noise can affect the shape of a single vehicle's trajectory when measurement sensors are affected by such phenomena. We could realize the necessity to use path reconstruction techniques such as the Kalman Filter [5], by providing some illustrative results on the performance of this specific tool. In this context, it is the intention of the authors to carry out the optimization of trajectories for a cooperative scenario where multiple vehicles need to take sudden maneuvering actions in order to avoid one or more obstacles, or at least reduce the impact of a hypothetical collision. By introducing noise in the measurement sensors and studying the influence on this sort of critical cooperative actions, it will be possible to analyze in detail how vehicles would react to such unpredictable phenomena in realistic situations. That is also one of the research lines the authors plan to investigate in the future.  
This work was supported by project grants MICINN/FEDER TEC2010-21405-C02-02/TCM (CALM) and 00002/CS/08 FORMA (_Fundacion Seneca- Ben Arabi Supercomputer RM_). It was also developed in the framework of '_Programa de Ayudas a Grupos de Excelencia de la Region de Murcia, Fundacion Seneca_'. [PERSON] also thanks the Spanish MICINN for a FPU (REF AP2008-02244) pre-doctoral fellowship.