Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation

TL;DR

This paper introduces a control method for managing roughening in the stochastic Kuramoto-Sivashinsky equation by splitting the problem and applying linear feedback control to stabilize and regulate the solution's roughness.

Contribution

It proposes a novel two-step control strategy combining stabilization and roughness regulation for stochastic PDEs, demonstrated on the Kuramoto-Sivashinsky equation.

Findings

Second moment evolves as a power-law before saturation

Control effectively stabilizes the zero solution

Both periodic and point actuated controls are effective

Abstract

We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.