Numerical Approximations for the Null Controllers of Structurally Damped Plate Dynamics

TL;DR

This paper develops and compares finite element and finite difference numerical schemes for approximating null controllers in structurally damped plate equations, demonstrating consistent asymptotic behavior and validating results through numerical experiments.

Contribution

It introduces fully-discrete FEM and FDM schemes for null controllability of damped plate equations and compares their asymptotic properties and numerical performance.

Findings

FEM and FDM null controllers share the same asymptotic minimal energy behavior.

Numerical experiments confirm theoretical asymptotics.

Both methods are effective for approximating null controllers.

Abstract

In this paper, we consider a structurally damped elastic equation under hinged boundary conditions. Fully-discrete numerical approximation schemes are generated for the null controllability of these parabolic-like PDEs. We mainly use finite element method (FEM) and finite difference method (FDM) approximations to show that the null controllers being approximated via FEM and FDM exhibit exactly the same asymptotics of the associated minimal energy function. For this, we appeal to the theory originally given by R. Triggiani [20] for construction of null controllers of ODE systems. These null controllers are also amenable to our numerical implementation in which we discuss the aspects of FEM and FDM numerical approximations and compare both methodologies. We justify our theoretical results with the numerical experiments given for both approximation schemes.