# A fully discrete numerical control method for the wave equation

## TL;DR

This paper introduces a fully discrete finite element method for controlling the wave equation, achieving optimal convergence rates under certain geometric conditions by designing stabilization terms to minimize computational error.

## Contribution

It develops a novel fully discrete scheme with stabilization for the wave equation control problem, ensuring optimal convergence under the geometric control condition.

## Key findings

- Achieves optimal convergence rate with respect to mesh parameter h.
- Designs stabilization terms to minimize computational error.
- Validates effectiveness under the geometric control condition.

## Abstract

We present a fully discrete finite element method for the interior null controllability problem subject to the wave equation. For the numerical scheme, piece-wise affine continuous elements in space and finite differences in time are considered. We show that if the sharp geometric control condition holds, our numerical scheme yields the optimal rate of convergence with respect to the space-time mesh parameter $h$. The approach is based on the design of stabilization terms for the discrete scheme with the goal of minimizing the computational error.