Boundary controllability for a degenerate beam equation

TL;DR

This paper investigates the boundary controllability of a degenerate beam equation, establishing existence, energy estimates, observability, and null controllability results for the system.

Contribution

It introduces a novel approach to boundary controllability for degenerate beam equations, including the use of transposition solutions and energy estimates.

Findings

Proved existence of solutions for the homogeneous problem

Established energy estimates and observability inequality

Demonstrated null controllability of the system

Abstract

The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control $f$ acts on the right end of it. As a first step we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them we prove an observability inequality and, using the notion of solution by transposition, we prove that the initial problem is null controllable.