Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance

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 $\mathit{\text{modified}}\mathit{\text{Euler}}\mathit{\text{equations}}\mathit{\text{for}}\mathit{\text{an}}\mathit{\text{inviscid}}\mathit{\text{fluid}}$, 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.