## 2 Background  
Efforts to reduce fuel consumption in automobiles, such as redesigning the internal combustion engine, reducing the mass and aerodynamic and rolling resistances [1] and using other propulsion technologies [2; 3; 4] have all been made. Today, optimization is present in the redesign of vehicle control systems [5] and other automotive parts [6] in order to obtain lower fuel consumption rates. There are several techniques utilized to optimize automobile power-train systems, such as, genetic algorithms (GA) [7], Simulated Annealing SA [8], Sequential Quadratic Programming (SQP) [9] and Dynamic Programming (DP) [10; 11; 12]. Several management strategies have been implemented for power-train control. These methods try to reduce the required energy. For example, different approaches tried to optimize hybrid vehicles operation. According to [13], global optimization is used when the optimization process uses a Hamilton-Jacobi-Bellman approach, such as DP. The results provided by DP can be considered as a benchmark for optimization [14; 15] because it finds the global minimum value. The methods based on Calculus of Variations obtain a local optimization result, for example, [PERSON]'s minimum principle. Other examples of optimization can be found in [16; 17; 18; 19]. In addition, other strategies have also been implemented. One of them is called Eco-driving. Eco-Driving can be described as the driving style followed by vehicle drivers to save fuel. This way of driving is based mainly on advance warnings and prediction, so the optimal average speed is selected at every instant and aggressive accelerations and decelerations are avoided by using traffic information and the geometric characteristics of the road ahead [20]. Classical Eco-driving tips have been widely distributed by car manufacturing companies, and some on-board devices can even collect data to monitor and evaluate the driving style. Eco-driving provides fuel savings from 15% [21] up to 25% [22]. In [23], the effect of driving style on fuel consumption and exhaust emissions is studied with the conclusion being reached that eco-driving can lead to 14% fuel savings but aggressive style increases consumption by 40%. Other authors present less positive figures, such as [24] with fuel savings of 3%-4%. Eco-driving strategies can be learnt by drivers following certain rules. However, observance of those rules is quite subjective, because it is difficult to establish specific limits. Some tests have been carried out to determine the influence of average speed on fuel consumption and travel time. In [25], two different roads were driven along using the same Eco-driving style. Three different average speeds were obtained. The travel duration time and fuel consumption were measured. The results show fuel reductions of around 27.6%, 10.8% and 10.9% when the travel times were increased by 7.8%, 6.25% and 7.5% respectively. The proportion of fuel consumption savings and extra time were 3.52, 1.73 and 1.46. It can be observed from these results that fuel savings and extra time do not vary in the same way. They depend on factors such as travel distance, maximum allowed speed and the digital map of the road. There are devices that show the driver the necessary instantaneous speed to arrive just-in-time at the destination [26]. These devices follow a reference travel time but fuel consumption is not considered. The driving style classification in [27] proved that, sometimes, on a given road, the total travel emissions using a low speed are higher than the total emissions when a higher average speed is used. Other more complex systems can minimize fuel consumption without modifying the travel duration time [28]. In [24], fuel savings percentages of around 3%-4% are mentioned. In [29], experimental tests have demonstrated that a truck can reach a fuel reduction of 3.5% without modifying its travel duration time over a road distance of 120 km. In automatic train control systems, similar concepts have been applied for optimize energy.