Stabilization of Non-Diagonal Infinite-Dimensional Systems with Delay Boundary Control

TL;DR

This paper introduces a novel boundary control method for stabilizing non-diagonal infinite-dimensional systems with delayed boundary input, combining backstepping and proportional control via the Artstein transformation.

Contribution

It presents the first integration of backstepping with direct-proportional control for delayed boundary stabilization of non-diagonal systems.

Findings

Successful stabilization of unstable system components

Effective handling of boundary control delay

Illustrative example demonstrating the method's applicability

Abstract

Here we deal with the stabilization problem of non-diagonal systems by boundary control. In the studied setting, the boundary control input is subject to a constant delay. We use the spectral decomposition method and split the system into two components: an unstable and a stable one. To stabilize the unstable part of the system, we connect, for the first time in the literature, the famous backstepping control design technique with the direct-proportional control design. More precisely, we construct a proportional open-loop stabilizer, then, by means of the Artstein transformation we close the loop. At the end of the paper, an example is provided in order to illustrate the acquired results.