Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance
For researchers in fluid dynamics and control, this provides a variational framework for obstacle avoidance in ideal flows, though the modification is incremental.
The paper extends optimal control of incompressible ideal flows to include obstacle avoidance by adding a barrier potential, resulting in modified Euler equations where the barrier term shifts effective pressure. Numerical results show localized flow deformation near obstacles.
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.