We present a new approach to the optimization of plasma scenarios in tokamaks. We formulate this task as an optimal control problem and use numerical methods for optimization problems with partial differential equation (PDE) constraints. The latter are discretized by linear finite elements and implicit Euler time stepping. Due to the free-boundary setting, we have introduced a new linearization of the non-linear equations that is consistent with the discretization. It is only this consistency, overlooked in previous approaches for the even simpler static setting, that guarantees that the method converges to the optimum. We give various numerical tests and simulation that illustrate our new method.
