Optimal design of equilibrium solutions of the Vlasov-Poisson system by an external electric field

A new optimization framework to design steady equilibrium solutions of the Vlasov-Poisson system by means of external electric fields is presented. This optimization framework requires the minimization of an ensemble functional with Tikhonov regularization of the control field under the differential constraint of a nonlinear elliptic equation that models equilibrium solutions of the Vlasov-Poisson system. Existence of optimal control fields and their characterization as solutions to first-order optimality conditions are discussed. Numerical approximations and optimization schemes are developed to validate the proposed framework.