Imperial College London

Professor Grigorios A. Pavliotis

Faculty of Natural SciencesDepartment of Mathematics

Professor of Applied Mathematics



+44 (0)20 7594 8564g.pavliotis Website




736aHuxley BuildingSouth Kensington Campus






BibTex format

author = {Tomlin, R and Gomes, SN and Pavliotis, G and Papageorgiou, D},
journal = {SIAM Journal on Applied Dynamical Systems},
title = {Optimal control of thin liquid films and transverse mode effects},
url = {},

RIS format (EndNote, RefMan)

AB - We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at anon-zero angle to the horizontal. Controls are applied via same-fluid blowing and suction throughthe substrate surface. We consider both overlying and hanging films, where the liquid lies above orbelow the substrate, respectively. We study the weakly nonlinear evolution of the system, which isgoverned by a forced Kuramoto–Sivashinsky equation in two space dimensions. The uncontrolledproblem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat filmsolution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes.We proceed with the assumption that we may actuate at every point on the substrate. The mainfocus is the optimal control problem, which we first study in the special case that the forcing mayonly vary in the spanwise direction. The structure of the Kuramoto–Sivashinsky equation allows theexplicit construction of optimal controls in this case using the classical theory of linear quadraticregulators. Such controls are employed to prevent the exponential growth of transverse waves inthe case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. Wethen consider the optimal control problem in full generality, and prove the existence of an optimalcontrol. For numerical simulations, we employ an iterative gradient descent algorithm. In the finalsection, we consider the effects of transverse mode forcing on the chaotic dynamics present in thestreamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearityallows us to reduce the average energy in solutions without directly forcing the linearly unstable dominant modes.
AU - Tomlin,R
AU - Gomes,SN
AU - Pavliotis,G
AU - Papageorgiou,D
SN - 1536-0040
TI - Optimal control of thin liquid films and transverse mode effects
T2 - SIAM Journal on Applied Dynamical Systems
UR -
ER -