Controlling nonlinear PDEs arising in fluid mechanics: open loop-, feedback control, and neural networks
Publication Date
July 30, 2026
Creator
Sattam Alrashidy
Abstract
This thesis addresses the control of systems governed by nonlinear partial differential equations, combining mathematical modelling, optimisation, and neural-network tools within a unified framework. The primary application is thin liquid films flowing over flexible surfaces, where the film and the underlying wall affect one another. A central question is how external actuation should be applied so that the film evolves in a controlled and predictable way. To study this, the thesis develops mathematical frameworks for PDE control together with numerical and neural-network approaches that support actuator placement and feedback design. The neural-network methods provide flexible representations for control strategies and help to incorporate the governing equations into the optimisation process. The work is organised in two parts: the first focuses on open-loop control of thin-film systems, and the second introduces actuator-design and feedback-control techniques using neural-network tools.
The first part of the thesis develops a PDE-constrained optimisation framework for open-loop control of thin-film flow over a flexible substrate. The model couples a nonlinear lubrication equation with an elastic-damping equation for the wall, capturing the interaction between the film and the flexible surface. The system allows for film rupture and coalescence and includes a global energy-dissipation law that reflects the physical structure of the problem. The control problem seeks a distributed forcing of the wall that drives the film towards a desired state. Optimality conditions are derived at the continuous level and solved numerically using finite-element discretisation and implicit-explicit time integration. The numerical simulations show how wall forcing can suppress instabilities, stabilise dewetting, and accelerate convergence towards stable film shapes. This formulation provides a foundation for PDE-constrained optimisation of thin-film systems.
The second part extends this framework to actuator design and feedback control. A one-shot optimisation formulation is introduced, in which actuator placement, control inputs, and state variables are optimised within a single PDE-constrained problem. This avoids the nested and adjoint structures of classical bilevel approaches. In the open-loop setting, physics-informed neural networks are used as a trial space for the thin-film dynamics, with actuator parameters trained alongside the control signal. Numerical experiments show that the framework identifies suitable actuator locations and control inputs, with finite-element simulations used for validation. In the feedback setting, the Hamilton-Jacobi-Bellman framework is extended to include actuator design parameters. The value function is represented by a neural network whose training jointly optimises the feedback law and the actuator configuration. This removes the need for explicit HJB solvers or precomputed datasets. Numerical experiments on the heat equation confirm that the learned controllers achieve performance comparable to Riccati-based feedback whilst enabling flexible actuator adaptation.
Overall, the thesis combines mathematical analysis, PDE modelling, and learning-based optimisation. It develops open-loop control and actuator-design tools for thin-film systems on flexible substrates and introduces a learning-based feedback framework in a simplified PDE setting. The feedback methodology is demonstrated for the heat equation, indicating a possible direction for future feedback control of thin-film flows.
Item Type
ethesis
Thesis Type
PhD
Supervisors
Kalise, Dante
Subjects (LC)
Associated Schools / Departments
UoN Repository URI
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