Optimal control of a semiclassical Boltzmann equation for charge transport in graphene

An ensemble optimal control problem governed by a semiclassical space-homogeneous Boltzmann equation for charge transport in graphene is formulated and analyzed. The control mechanism is an external time-dependent electric field having the purpose of driving the material density of electrons in graphene to follow a desired trajectory. For this purpose, an ensemble cost functional with quadratic H1 costs is considered. Well-posedness of the control problem is proved, and the characterization of optimal controls by an optimality system is discussed. This system is approximated by a discontinuous Galerkin scheme and solved using a nonlinear conjugate gradient method. Results of numerical experiments are presented that demonstrate the ability of the proposed control framework to design control fields.