高温等离子体的最优控制
Optimal Control of a Hot Plasma
摘要：
无碰撞等离子体的时间演化由相对论性的Vlasov-Maxwell系统来模拟，该系统耦合了Vlasov方程（输运方程）和电动力学的麦克斯韦方程组。我们考虑的情况下，等离子体位于一个有界的容器$Ω\subset\mathbb R^3$，例如聚变反应堆。此外，存在外部电流，通常在容器的外部，如果适当地调节，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$μ$（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保持撞击$\partialΩ$的粒子数量尽可能小（因为它们会损坏反应堆壁），同时控制成本不应太详尽（以确保效率）。这导致了一个最小化问题的PDE约束。对此问题进行了详细的分析。特别地，我们证明了极小值的存在性，并建立了一种方法来获得一阶最优性条件。
英文摘要：
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 $Ω\subset\mathbb R^3$, 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, that are placed in space, via given matrix-valued functions $\varepsilon$ (the permittivity) and $μ$ (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting $\partialΩ$ 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.
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