Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Sensing

TL;DR

This paper develops adaptive boundary control strategies for the Kuramoto-Sivashinsky equation with intermittent sensing, ensuring stability and boundedness despite unknown, space-dependent destabilizing coefficients and partial state measurements.

Contribution

It introduces adaptive boundary controllers for the KS equation under intermittent sensing, handling unknown perturbations and guaranteeing stability or boundedness.

Findings

Input-to-state stability with known perturbation bounds.

Global uniform ultimate boundedness with unknown perturbation size.

Convergence to small neighborhoods with full state measurement.

Abstract

We study in this paper boundary stabilization, in the L2 sense, of the perturbed Kuramoto-Sivashinsky (KS) equation subject to intermittent sensing. We assume that we measure the state on a given spatial subdomain during certain time intervals, while we measure the state on the remaining spatial subdomain during the remaining time intervals. We assign a feedback law at the boundary of the spatial domain and force to zero the value of the state at the junction of the two subdomains. Throughout the study, the equation's destabilizing coefficient is assumed to be unknown and possibly space dependent but bounded. As a result, adaptive boundary controllers are designed under different assumptions on the perturbation. In particular, we guarantee input-to-state stability (ISS) when an upperbound on the perturbation's size is known. Otherwise, only global uniform ultimate boundedness (GUUB) is guaranteed. In contrast, when the state is measured at every spatial point all the time (full state measurement), convergence to an arbitrarily-small neighborhood of the origin is guaranteed, even if the perturbation's maximal size is unknown. Numerical simulations are performed to illustrate our results.