Point-actuated feedback control of multidimensional interfaces

Ruben J Tomlin,Susana N Gomes

doi:10.1093/imamat/hxz031

Abstract

AbstractWe consider the application of feedback control strategies with point actuators to multidimensional evolving interfaces in order to stabilize desired states. We take a Kuramoto–Sivashinsky equation as a test case; this equation arises in the study of thin liquid films, exhibiting a wide range of dynamics in different parameter regimes, including unbounded growth and full spatiotemporal chaos. The controls correspond physically to mass-flux actuators located in the substrate on which the liquid film lies. In the case of partial state observability, we utilize a proportional control strategy where forcing at a point depends only on the local observation. We find that point-actuated controls may inhibit unbounded growth of a solution, if the actuators are sufficient in number and in strength, and can exponentially stabilize the desired state. We investigate actuator arrangements, and find that the equidistant case is the most favourable for control performance, with a large drop in effectiveness for poorly arranged actuators. Proportional controls are also used to synchronize two chaotic solutions. When the interface is fully observable, we construct model-based controls using the linearization of the governing equation. These improve on proportional controls and are applied to stabilize non-trivial steady and travelling wave solutions.

Highlights

The study of evolving interfaces is at the core of many areas of applied mathematics, ranging from crystal growth problems in chemistry (Langer, 1980; Kobayashi, 1993; Pimpinelli & Villain, 1999) to the study of flame front propagation in combustion theory (Michelson & Sivashinsky, 1977; Sivashinsky, 1977, 1980)
It is often challenging to isolate the interfacial dynamics while still retaining all desired physical effects, and in many cases it is found that the obtained low-dimensional models only replicate the true dynamics well in restricted parameter regimes
The stochastic Kardar–Parisi–Zhang equation is used as a continuum model for the interfacial dynamics of crystal growth via the artificial deposition process known as molecular-beam epitaxy (Pimpinelli & Villain, 1999)

Summary

Introduction

The study of evolving interfaces is at the core of many areas of applied mathematics, ranging from crystal growth problems in chemistry (Langer, 1980; Kobayashi, 1993; Pimpinelli & Villain, 1999) to the study of flame front propagation in combustion theory (Michelson & Sivashinsky, 1977; Sivashinsky, 1977, 1980). It is often challenging to isolate the interfacial dynamics while still retaining all desired physical effects, and in many cases it is found that the obtained low-dimensional models only replicate the true dynamics well in restricted parameter regimes This pitfall is balanced by the relative simplicity of the interface evolution equations along with a large decrease in computational complexity for numerical simulations. Cooling and coating processes involving thin film flows arise in microfluidic applications; for cooling, waviness of fluid interfaces is desirable as it improves heat transfer (Miyara, 1999; Serifi et al, 2004), whereas for coating, a flat interface is sought. It may be the case that such a simplification overlooks important instabilities or mechanisms that are only observed from the full 3D formulation of the original multiphase problem, e.g. Rayleigh–Taylor instabilities or electrostatically induced instabilities in liquid films (Tomlin et al, 2017, 2019)

Objectives

Methods

Conclusion