In this paper, we analyze first-order methods to find a KKT point of the nonlinear optimization problems arising in Model Predictive Control (MPC). The methods are based on a projected gradient and constraint linearization approach, that is, every iteration is a gradient step, projected onto a linearization of the constraints around the current iterate. We introduce an approach that uses a simple ℓ_p merit function, which has the computational advantage of not requiring any estimate of the dual variables and keeping the penalty parameter bounded. We then prove global convergence of the proposed method to a KKT point of the nonlinear problem. The first-order methods can be readily implemented in practice via the novel tool FalcOpt. The performance is then illustrated on numerical examples and compared with conventional methods.