Analysis of Robin-boundary control for the Boussinesq equations

TL;DR

This paper investigates an optimal boundary control problem for the Boussinesq equations, establishing well-posedness, deriving optimality conditions, and performing numerical analysis with error estimates and validation.

Contribution

It introduces a comprehensive framework for boundary control of Boussinesq equations, including theoretical derivations and numerical validation with error estimates.

Findings

Well-posedness of the control problem established

Explicit optimality conditions derived

Numerical scheme validated with error estimates

Abstract

In this paper, we study an optimal boundary control problem for the Boussinesq equations, which couple the time-dependent Navier-Stokes system with a heat equation, where the control enters through a Robin boundary condition on temperature. We begin by establishing the well-posedness of the optimization problem via a variational framework. We then derive both first- and second-order optimality conditions, including explicit characterizations of the adjoint state and the optimal control. Next, we perform a detailed numerical analysis of a fully discrete scheme: using finite elements in space and a semi-implicit scheme in time, combined with variational discretization for the control. We present rigorous a prior error estimates for the state, adjoint state, and control variables. Numerical experiments are provided to validate our theoretical results.