SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2895-2929, January 2020.
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov--Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma is located in a bounded container $\Omega\subset\mathbb R^3$, for example a fusion reactor. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We model objects, which are placed in space, via given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting $\partial\Omega$ as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimizing problem with a PDE constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions.

中文翻译：

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容器中的热等离子体-最佳控制问题

SIAM数学分析杂志，第52卷，第3期，第2895-2929页，2020年1月。
相对论的Vlasov-Maxwell系统对无碰撞等离子体的时间演化进行建模，该系统将Vlasov方程（传输方程）与电动力学的Maxwell方程耦合。我们考虑等离子体位于有界容器$ \ Omega \ subset \ mathbb R ^ 3 $中的情况，例如聚变反应堆。此外，如果适当地调节，通常在容器的外部存在外部电流，其可以用作等离子体的控制。我们通过给定的矩阵值函数$ \ varepsilon $（介电常数）和$ \ mu $（磁导率）对放置在空间中的对象进行建模。聚变等离子体物理学的一个典型目标是使击中$ \ partial \ Omega $的粒子数量尽可能小（因为它们会损坏反应器壁），而控制成本不应过于详尽（以确保效率）。这导致PDE约束最小化的问题。将详细分析此问题。特别是，我们证明了极小值的存在，并建立了导出一阶最优条件的方法。