# Stability and Controllability results for a Timoshenko system

## TL;DR

This paper investigates the stability and controllability of a one-dimensional Timoshenko system, demonstrating polynomial energy decay under fractional damping and establishing exact controllability using boundary controls.

## Contribution

It provides new insights into the stability decay rates depending on damping coefficients and proves exact controllability with boundary controls under specific conditions.

## Key findings

- System is strongly stable but not uniformly stable.
- Energy decay rate depends on PDE coefficients and fractional damping order.
- System is exactly controllable with boundary control in finite time.

## Abstract

In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet-Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.