Optimal control of thin-film flow on a flexible topography
File(s)
Accepted version
Author(s)
Alrashidy, Sattam
Kalogirou, Anna
Kalise, Dante
Van der Zee, Kristoffer
Type
Journal Article
Abstract
This work presents a mathematical model for the optimal control of thin-film flows over a flexible substrate influenced by an external force. The objective is to find the optimal distributed force acting on the topography that minimises the differences between actual and desired thin-film profiles. A nonlinear lubrication equation governing the fluid dynamics and appropriate functional settings for this model are presented. It is also shown that this system satisfies a global energy-dissipation law for a suitable energy functional. Optimality conditions are derived for the solution of the minimisation problem of a specified cost function across a time horizon. These conditions are formulated at a continuous level as system of coupled, forward-backward PDEs, which are subsequently discretised for numerical investigation. To ensure computational efficiency and stability, first-order Implicit-Explicit (IMEX) time-stepping schemes are employed to handle the nonlinearities in the model, and a reduced gradient descent algorithm is applied to obtain a numerical approximation of the optimal control signal. Numerical results illustrate that controlling the thin film, even during rupture, achieves a precise film profile. This control strategy accelerates convergence towards a steady state, reduces instabilities, stabilises dewetting processes, and meets the desired profile specifications.
Date Issued
2026-08-01
Date Acceptance
2026-03-31
Citation
SIAM Journal on Applied Mathematics, 2026, 86 (4), pp.1807-1835
ISSN
0036-1399
Publisher
Society for Industrial and Applied Mathematics
Start Page
1807
End Page
1835
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
86
Issue
4
Copyright Statement
Copyright © 2026 Society for Industrial and Applied Mathematics. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2026-07-21