Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Data Availability: With Proofs

TL;DR

This paper develops boundary control strategies for the Kuramoto-Sivashinsky equation to achieve stabilization despite intermittent measurements, using boundary and interior point controls with stability analysis.

Contribution

It introduces two novel boundary controllers for the nonlinear Kuramoto-Sivashinsky equation under intermittent data, with stability proofs and control design based on measurements at boundary and interior points.

Findings

Stability of the closed-loop system is proven for both controllers.

Effective control is achieved with measurements only at boundary and interior points.

The approach handles intermittent measurement scenarios successfully.

Abstract

In this paper, two boundary controllers are proposed to stabilize the origin of the nonlinear Kuramoto-Sivashinsky equation under intermittent measurements. More precisely, the spatial domain is divided into two sub-domains. The state of the system on the first sub-domain is measured along a given interval of time, and the state on the remaining sub-domain is measured along another interval of time. Under the proposed sensing scenario, we control the considered equation by designing the value of the state at three isolated spatial points, the two extremities of the spatial domain plus one inside point. Furthermore, we impose a null value for the spatial gradient of the state at these three locations. Under such a control loop, we propose two types of controllers and we analyze the stability of the resulting closed-loop system in each case. The paper is concluded with some discussions and future works.