不可压缩理想流体障碍回避最优控制
Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance
AI总结：
本文提出一种变分障碍回避形式化方法，通过引入屏障势函数修改欧拉方程，研究障碍回避对流体动力学的影响，展示屏障项对流体局部变形的作用。
摘要：
本文展示了一种变分障碍回避形式化方法，通过在关联最优控制泛函中引入屏障型势函数，修改欧拉方程。这导致了无粘性流体的修正欧拉方程，其中屏障项作用于拉格朗日配置，并在欧拉描述中表现为有效压力的偏移。我们还展示了减少的欧拉动力学的数值示例，表明屏障项在障碍区域诱导了流体的局部变形，与其作为障碍回避惩罚的作用一致。
英文摘要：
It was shown in \cite{bloch2000optimal} that an optimal control formulation for incompressible ideal fluid flow yields Euler's equations. In this paper, we consider a variational obstacle-avoidance formulation for incompressible ideal flows by introducing a barrier-type potential in the associated optimal control functional. This leads to \textit{modified Euler equations for an inviscid fluid}, in which the barrier term acts on the Lagrangian configuration and appears in the Eulerian description as a shift in the effective pressure. We also present a numerical illustration of the reduced Eulerian dynamics, showing that the barrier term induces a localized deformation of the flow near the obstacle region, consistent with its role as an obstacle-avoidance penalization.
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