# Carleman estimate for an adjoint of a damped beam equation and an application to null controllability

## TL;DR

This paper develops a new Carleman estimate for a damped beam equation's adjoint and uses it to explicitly construct control functions that steer the system to a null state, advancing control theory for PDEs.

## Contribution

It introduces a novel Carleman estimate for the adjoint of a damped beam equation and applies duality to explicitly determine control functions for null controllability.

## Key findings

- Derived a new Carleman estimate for the adjoint equation.
- Explicitly constructed control functions for null controllability.
- Validated the control approach for a one-dimensional damped beam model.

## Abstract

In this article we consider a control problem of a linear Euler-Bernoulli damped beam equation with potential in dimension one with periodic boundary conditions. We derive a new Carleman estimate for an adjoint of the equation under consideration. Then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state.