The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated needs the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this paper, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides a feedback on the equation of motion based on an instantaneous prediction of the discretized system. This permits to directly embed the minimization of a given cost functional into the particle interactions of the corresponding Vlasov model. The numerical results demonstrate the validity of our control approach and the capability of an external magnetic field, even if in a simplified setting, to lead the plasma far from the boundaries.

翻译：磁约束聚变的基本原理是通过将高温等离子体约束在装置内部，使氘核和氚核结合释放能量。为保护装置壁面免遭损坏，需将等离子体与器壁隔离，而外部磁场的作用正是实现这一目标。本文从数值模拟角度应对这一挑战，提出一种瞬时控制数学方法，使等离子体被引导至指定空间区域。在建模层面，我们聚焦于有界域内考虑自感应电场与强外部磁场的Vlasov方程。该控制策略的核心特征在于：基于离散化系统的瞬时预测，对运动方程施加反馈作用。这使得给定代价函数的最小化能够直接嵌入相应Vlasov模型中的粒子相互作用过程。数值结果验证了该控制方法的有效性，表明即使采用简化模型，外部磁场仍具备将等离子体引导至远离边界区域的能力。