In this project, an attempt is made to use model based optimal control for CERN's AWAKE electron beam line trajectory correction problem. CERN is constantly searching for advanced control methods for the new accelerators and this model based approach using an Linear Quadratic Regulator (LQR) in this manner has not been attempted before. The report therefore, begins with a description of CERN and the main accelerators and its experiments. The AWAKE experiment is described in detail, with focus on the electron beam line. Then, a thorough charged particle modelling chapter is presented with derivations through magnetic effects on the system all the way to linear equations of motion. Based on this, extensive state of the art research was done to analyse the control methods currently applied at CERN. Afterwards, the requirements and the control objective is defined. The system is then modelled in a simple approach to prove that model based control approaches can work on ultra-relativistic time-like systems. The model is then put into state space form and discretized. This yielded a bilinear system which is then controlled with an LQR controller. Simulations are then created which are based on a Reinforcement Learning simulation environment, which is modified to use the encoded physical properties of the system elements for the LQR simulation. In the end, the results show that the LQR feedback controller managed to drive the beam trajectory towards the reference. However, due to lack of access to physical hardware, it is difficult to assess real world performance.