Optimal design of equilibrium solutions of the Vlasov-Poisson system by an external electric field (Q6669439)

A new optimization framework is presented for designing steady equilibrium solutions of the Vlasov-Poisson system using external electric fields. This framework involves minimizing an ensemble functional with Tikhonov regularization of the control field, subject to the differential constraint of a nonlinear elliptic equation that represents the equilibrium solutions of the Vlasov-Poisson system. The 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.