Numerical controllability of the wave equation through primal methods and Carleman estimates

TL;DR

This paper presents a direct primal approach using Carleman estimates for the numerical computation of boundary controls in the 1D wave equation, avoiding duality methods and demonstrating convergence through finite element approximations.

Contribution

It introduces a novel primal control method based on Carleman estimates and establishes convergence of finite element approximations for the control problem.

Findings

Finite element method approximations converge strongly.

Numerical experiments validate the theoretical results.

The approach avoids duality arguments in control computation.

Abstract

This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute an approximation of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not use in this work duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and of the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.