On the Computation of Hierarchical Control results for One-Dimensional Transmission Line

TL;DR

This paper explores hierarchical controllability in a one-dimensional wave equation with a moving boundary, introducing a numerical approach combining FEM and FDM to evaluate control strategies.

Contribution

It introduces a hierarchical control framework with a Stackelberg strategy for a wave equation with moving boundary, and develops a discretization algorithm for numerical analysis.

Findings

Algorithm effectively evaluates control strategies.

Numerical results demonstrate the approach's efficiency.

Software implementations confirm practical applicability.

Abstract

In this paper, motivated by a physics problem, we investigate some numerical and computational aspects for the problem of hierarchical controllability in a one-dimensional wave equation in domains with a moving boundary. Some controls act in part of the boundary and define a strategy of equilibrium between them, considering a leader control and a follower. Thus, we introduced the concept of hierarchical control to solve the problem and mapped the Stackelberg Strategy between these controls. A total discretization of the problem is presented for a numerical evaluation in spaces of finite dimension, an algorithm for evaluation of the problem is presented as the combination of finite element method (FEM) and finite difference method (FDM). The algorithm efficiency and computational results are illustrated for some experiments using the softwares Freefem++ and MatLab.