## 1 Introduction
The model proposed by [PERSON] and [PERSON] [13] in 1994 calculates pollutant raw emissions of a spark ignition engine and takes into account numerous parameters such as throttle angle, spark advance and fuel/air ratio. They minimize step by step a criteria taking into account fuel consumption and raw pollutant emissions over the first elementary profile of speed in the European driving cycle. Successive studies have then explored this topic in the literature, studying the influence of other variables, reducing the complexity of models or using techniques such as artificial neural networks to formulate accurate models from a limited number of experimental data [14, 15]. In 2012, [PERSON] et al. [16] developed a model that took into account the same variables as [PERSON]'s, but introduced the valve timing (synchronization of cylinder opening and closing with piston movement) and a three way catalyst model. The integrated model comprised 65 state variables and many non linearities. A reduction of the model made it possible to neglect some of these variables depending on the pollutant considered. According to this work, at least 7 state variables were mandatory in order to simulate exhaust pollutants with a good precision. However, this is far from being compatible with optimal control algorithm. Consequently, another approach has to be considered. The main disadvantage of empirical hypothesis is that the models obtained are valid for a specific engine. Indeed, they are difficult to adapt to other engines because they do not take into account specificity such as combustion chamber parameters (geometry, surface/volume ratio, turbulence intensity). It is, therefore, necessary to calibrate the models for the engine under study [17]. The control parameters selected for our study are intake manifold pressure, fuel/air ratio, spark advance and engine speed. Our simulation framework will include the following \"main\" parts:  
* A simple backward model of the drivetrain (i.e., from drive cycle to engine) as described by [18] to calculate at each time step the engine speed and torque. * A Mean Value Engine Model inspired from a [PERSON] description [12] and adapted from [PERSON] et al. [19] to include the effect of control parameters over overall efficiency. * Simple forms to describe the behavior of emissions with a semi empirical model based on experimental data. A catalytic converter model describing its thermal and efficiency behaviors with only one spatial discretization as proposed by [PERSON] et al. [6]. * A dynamic programming algorithm to calculate the optimal control parameters of the engine over an entire drive cycle by minimizing a weighted sum of pollutant emissions and fuel consumption. This implementation was derived from [20]. The main objective of this work is to explore trade-offs between fuel consumption and pollutant emissions while adopting a systemic approach. By that, we mean considering the entire vehicle as a whole system, starting from a drive cycle up to the after treatment system, and taking into account drive train, engine and so on. The models are described in Section 2, followed by the results in Section 3. The originality of this approach will be further discussed in Section 4.