# Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier-Stokes-Voigt equations

## TL;DR

This paper develops and analyzes a fully discrete discontinuous Galerkin scheme combined with finite element methods for optimal control of 3D Navier-Stokes-Voigt equations, providing error estimates for control approximations.

## Contribution

It introduces a novel fully discrete scheme for the optimal control problem of Navier-Stokes-Voigt equations with proven error bounds.

## Key findings

- Error estimates of order $O(\sqrt{ au}+h)$ for control approximations.
- Convergence of the discrete controls to the continuous optimal controls.
- Validation of the scheme's effectiveness for 3D Navier-Stokes-Voigt control problems.

## Abstract

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier-Stokes-Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order $O(\sqrt{\tau}+h)$, where $\tau$ and $h$ are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.