# Singular asymptotic expansion of the exact control for a linear model of the Rayleigh beam

## TL;DR

This paper derives a precise asymptotic expansion of the minimal control for a Petrowsky type beam equation as a small parameter approaches zero, revealing the boundary layer effects and connecting to classical wave control results.

## Contribution

It provides a rigorous second order asymptotic expansion of the control for the Petrowsky equation, including boundary layer analysis, extending classical control results.

## Key findings

- Boundary layer of size √ε at the extremities.
- Leading control term is a null Dirichlet control for the wave limit.
- Numerical experiments validate the asymptotic analysis.

## Abstract

The Petrowsky type equation $y_{tt}^\eps+\eps y_{xxxx}^\eps - y_{xx}^\eps=0$, $\eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\eps}$ occurring at the extremities, these boundary controls get singular as $\eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^\eps$ then derive a rigorous second order asymptotic expansion of the control of minimal $L^2-$norm, with respect to the parameter $\eps$. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.