Controlling a Vlasov–Poisson Plasma by a Particle-in-Cell Method Based on a Monte Carlo Framework

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-07-10 DOI:10.1137/23m1563852

Jan Bartsch, Patrik Knopf, Stefania Scheurer, Jörg Weber

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 1977-2011, August 2024.
Abstract. The Vlasov–Poisson system describes the time evolution of a plasma in the so-called collisionless regime. The investigation of a high-temperature plasma that is influenced by an exterior magnetic field is one of the most significant aspects of thermonuclear fusion research. In this paper, we formulate and analyze a kinetic optimal control problem for the Vlasov–Poisson system where the control is represented by an external magnetic field. The main goal of such optimal control problems is to confine the plasma to a certain region in phase space. We first investigate the optimal control problem in terms of mathematical analysis, i.e., we show the existence of at least one global minimizer and rigorously derive a first-order necessary optimality condition for local minimizers by the adjoint approach. Then we build a Monte Carlo framework to solve the state equations as well as the adjoint equations by means of a particle-in-cell method, and we apply a nonlinear conjugate gradient method to solve the optimization problem. Eventually, we present numerical experiments that successfully validate our optimization framework.

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用基于蒙特卡洛框架的胞内粒子法控制弗拉索夫-泊松等离子体

SIAM 控制与优化期刊》，第 62 卷第 4 期，第 1977-2011 页，2024 年 8 月。摘要Vlasov-Poisson 系统描述了等离子体在所谓无碰撞状态下的时间演化。研究受外部磁场影响的高温等离子体是热核聚变研究中最重要的方面之一。在本文中，我们提出并分析了一个由外部磁场控制的 Vlasov-Poisson 系统的动力学最优控制问题。此类最优控制问题的主要目标是将等离子体限制在相空间的某个区域内。我们首先从数学分析的角度研究了最优控制问题，即证明至少存在一个全局最小值，并通过邻接法严格推导出局部最小值的一阶必要最优条件。然后，我们建立了一个蒙特卡罗框架，通过粒子入胞法求解状态方程和邻接方程，并应用非线性共轭梯度法求解优化问题。最后，我们通过数值实验成功验证了我们的优化框架。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.