In this thesis, we present novel model predictive control (MPC) formulations based on a convex open-loop optimal control problem to tackle the problem setup of trajectory tracking and path following as well as the control of systems with unknown system dynamic. In particular, we consider the framework of relaxed barrier function based MPC (rb MPC). We extend the existing stability theory to the trajectory tracking and the path following problem. We establish important system theoretic properties like closed-loop stability and exact constraint satisfaction under suitable assumptions. Moreover, we evaluate the developed MPC algorithms in the area of automated driving in simulations as well as in a real-world driving scenario. Further, we consider the control of completely unknown systems based on online optimization. We divide the overall problem into the design of an estimation algorithm and a control algorithm. The control algorithm is a model-independent receding horizon control algorithm in which important system theoretic properties like convergence to the origin are guaranteed without the knowledge of the true system parameters. The estimation and control algorithm are combined together and convergence to the origin of the closed-loop system for fully unknown linear time-invariant discrete-time systems is shown.