Point-actuated feedback control of multidimensional interfaces

Abstract

We consider the application of feedback control strategies with point actuators to multidimensional evolv-ing interfaces in order to stabilise 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 differentparameter regimes, including unbounded growth and full spatiotemporal chaos. The controls correspondphysically to mass-flux actuators located in the substrate on which the liquid film lies. In the case of par-tial state observability, we utilise a proportional control strategy where forcing at a point depends only onthe 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 stabilise the desired state.We investigate actuator arrangements, and find that the equidistant case is the most favourable for con-trol performance, with a large drop in effectiveness for poorly arranged actuators. Proportional controlsare also used to synchronise two chaotic solutions. When the interface is fully observable, we constructmodel-based controls using the linearisation of the governing equation. These improve on proportionalcontrols, and are applied to stabilise non-trivial steady and travelling wave solutions.