Optimal control of viscous fingering

The paper considers the problem of optimally filling a Hele-Shaw cell. The system is subject to viscous fingering effect. It is shown that, despite the threshold terms appearing on the right-hand side of the governing equations, the dynamics can be rewritten using several prime integrals. This allows reforming optimal control problems for the Fourier modes describing the fluid interface into smooth optimization problems, in the sense of Gateaux derivative. Some numerical experiments illustrate the advantages of using the optimal solutions obtained using this reformulation instead of the currently known time-dependent injection rates.