Fusion energy offers the potential for the generation of clean, safe, and nearly inexhaustible energy. While notable progress has been made in recent years, significant challenges persist in achieving net energy gain. Improving plasma confinement and stability stands as a crucial task in this regard and requires optimization and control of the plasma system. In this work, we deploy a PDE-constrained optimization formulation that uses a kinetic description for plasma dynamics as the constraint. This is to optimize, over all possible controllable external electric fields, the stability of the plasma dynamics under the condition that the Vlasov--Poisson (VP) equation is satisfied. For computing the functional derivative with respect to the external field in the optimization updates, the adjoint equation is derived. Furthermore, in the discrete setting, where we employ the semi-Lagrangian method as the forward solver, we also explicitly formulate the corresponding adjoint solver and the gradient as the discrete analogy to the adjoint equation and the Frechet derivative. A distinct feature we observed of this constrained optimization is the complex landscape of the objective function and the existence of numerous local minima, largely due to the hyperbolic nature of the VP system. To overcome this issue, we utilize a gradient-accelerated genetic algorithm, leveraging the advantages of the genetic algorithm's exploration feature to cover a broader search of the solution space and the fast local convergence aided by the gradient information. We show that our algorithm obtains good electric fields that are able to maintain a prescribed profile in a beam shaping problem and uses nonlinear effects to suppress plasma instability in a two-stream configuration.

翻译：聚变能源具有产生清洁、安全且近乎无限能源的潜力。尽管近年来取得了显著进展，但在实现净能量增益方面仍存在重大挑战。改善等离子体约束与稳定性是此领域的关键任务，需要对等离子体系统进行优化与控制。本研究提出了一种基于偏微分方程约束的优化框架，以等离子体动力学的动理学描述作为约束条件。该框架旨在所有可控外部电场中，寻找满足Vlasov-Poisson（VP）方程条件下使等离子体动力学稳定性最优的电场。为计算优化迭代中对外部电场的泛函导数，我们推导了伴随方程。此外，在采用半拉格朗日方法作为正向求解器的离散框架中，我们明确构建了相应的伴随求解器，并将梯度定义为伴随方程与弗莱歇导数的离散类比。该约束优化的一个显著特征是目标函数的复杂景观及大量局部极小值的出现，这主要源于VP系统的双曲特性。为克服这一问题，我们采用了梯度加速遗传算法，既利用遗传算法探索特性覆盖更广的解空间搜索范围，又借助梯度信息实现快速局部收敛。研究表明，该算法在束流整形问题中可获取能够维持预设轮廓的优质电场，并通过非线性效应在双流构型中有效抑制等离子体不稳定性。