Analysis and Optimal Design of Equilibria in the Vlasov-Poisson System

Alfio Borzì; Gennaro Infante; Giovanni Mascali; Riccardo Scida

Analysis and Optimal Design of Equilibria in the Vlasov-Poisson System

Optimization and Control 2026-07-28 v1

Authors: Alfio Borzì , Gennaro Infante , Giovanni Mascali , Riccardo Scida

Abstract

The optimal design of equilibrium particle distributions generated by external electric fields is investigated in the framework of stationary Maxwellian equilibria of the Vlasov-Poisson system. The resulting normalized Poisson-Boltzmann equation is analyzed, and existence and uniqueness of the self-consistent equilibrium potential are established by two complementary approaches. A Schauder fixed-point argument based on uniform estimates for the normalized nonlinear source and a variational approach based on a strictly convex free-energy functional are developed. Building upon these analytical results, a unified optimal design framework is formulated for equilibrium densities. Valley-potential, quadratic tracking, and Kullback-Leibler design criteria are treated within the same optimization framework. First-order optimality conditions are derived, and a projected nonlinear conjugate-gradient algorithm is proposed for the numerical solution of the resulting optimization problems.

Keywords

Vlasov-Poisson system optimal design Poisson-Boltzmann equation Schauder fixed-point theorem variational method conjugate-gradient algorithm

Cite

@article{arxiv.2607.25698,
  title  = {Analysis and Optimal Design of Equilibria in the Vlasov-Poisson System},
  author = {Alfio Borzì and Gennaro Infante and Giovanni Mascali and Riccardo Scida},
  journal= {arXiv preprint arXiv:2607.25698},
  year   = {2026}
}