通过外部电场实现 Vlasov-Poisson 系统平衡解的最优设计
Optimal design of equilibrium solutions of the Vlasov-Poisson system by an external electric field
AI总结：
本文提出一种通过外部电场设计 Vlasov-Poisson 系统稳态平衡解的优化框架，在非线性椭圆方程约束下最小化带 Tikhonov 正则化的系综泛函，证明了最优控制场的存在性并开发数值方案验证。
摘要：
本文提出了一种新的优化框架，通过外部电场设计 Vlasov-Poisson 系统的稳态平衡解。该框架要求在模拟 Vlasov-Poisson 系统平衡解的非线性椭圆方程微分约束下，最小化带有控制场 Tikhonov 正则化的系综泛函。讨论了最优控制场的存在性及其作为一阶最优性条件解的表征。开发了数值近似和优化方案以验证所提框架。
英文摘要：
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.
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