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Analysis of Robin-boundary control for the Boussinesq equations

Wei Gong, Dongdong Liang, Xianbing Luo, Changlun Ye

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.

Analysis of Robin-boundary control for the Boussinesq equations

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.
Paper Structure (15 sections, 41 theorems, 229 equations, 1 figure)

This paper contains 15 sections, 41 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal control problem
  4. The Boussinesq equations
  5. The optimality conditions
  6. Regularity of the optimal solutions
  7. Numerical approximation of the optimal control problem
  8. Spatial discretization
  9. The fully-discrete optimal control problem
  10. Analysis of the fully-discrete state equation
  11. Analysis of the discrete adjoint equation
  12. Analysis of the discrete control problem
  13. Numerical Experiments
  14. Fundamental Lemmas
  15. Supplementary Lemma

Key Result

Lemma 3.1

The problem intr:Rb_state admits a unique weak solution $(\mathbf y, \theta )\in \mathbb W$ such that

Figures (1)

  • Figure 1: The magnitude of the target velocity $\mathbf y_d$

Theorems & Definitions (80)

  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 3.1
  • ...and 70 more