# Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems with Delay Boundary Control

## TL;DR

This paper presents a novel boundary feedback control method that stabilizes diagonal infinite-dimensional systems with delayed boundary inputs, ensuring exponential ISS and stability of interconnected systems.

## Contribution

It introduces a finite-dimensional truncation approach combined with Lyapunov analysis to stabilize infinite-dimensional systems with delayed boundary control.

## Key findings

- Finite-dimensional delay controller stabilizes the original system.
- Closed-loop system exhibits exponential Input-to-State Stability.
- Small gain condition guarantees stability of IDS-ODE interconnection.

## Abstract

This paper studies the boundary feedback stabilization of a class of diagonal infinite-dimensional boundary control systems. In the studied setting, the boundary control input is subject to a constant delay while the open loop system might exhibit a finite number of unstable modes. The proposed control design strategy consists in two main steps. First, a finite-dimensional subsystem is obtained by truncation of the original Infinite-Dimensional System (IDS) via modal decomposition. It includes the unstable components of the infinite-dimensional system and allows the design of a finite-dimensional delay controller by means of the Artstein transformation and the pole-shifting theorem. Second, it is shown via the selection of an adequate Lyapunov function that 1) the finite-dimensional delay controller successfully stabilizes the original infinite-dimensional system; 2) the closed-loop system is exponentially Input-to-State Stable (ISS) with respect to distributed disturbances. Finally, the obtained ISS property is used to derive a small gain condition ensuring the stability of an IDS-ODE interconnection.