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A posteriori error estimates for a bang-bang optimal control problem

Francisco Fuica

TL;DR

The work develops a residual-type a posteriori error estimator for a PDE-constrained bang-bang optimal control problem analyzed without Tikhonov regularization, using variational discretization for the state and adjoint while keeping the control in the continuous admissible set. The estimator decomposes into state and adjoint components and is shown to be reliable and locally efficient on convex domains; a growth condition on the adjoint state yields a robust bound on the control and state errors in terms of the estimators, with a simplified form when the growth exponent equals 1. An adaptive finite element loop driven by the total estimator achieves optimal experimental convergence rates for all variables, and numerical experiments on convex and non-convex domains (including cases with unbounded forcing in the adjoint) validate the theory and demonstrate practical performance. The results provide a rigorous, practical framework for AFEM-based solution of bang-bang OCPs without regularization, including scenarios with unknown exact solutions and varying adjoint-growth behavior. The approach is well-suited for applications requiring accurate error control and efficient mesh refinement in PDE-constrained optimization with bang-bang controls.

Abstract

We propose and analyze a posteriori error estimates for a control-constrained optimal control problem with bang-bang solutions. We consider a solution strategy based on the variational approach, where the control variable is not discretized; no Tikhonov regularization is made. We design, for the proposed scheme, a residual-type a posteriori error estimator that can be decomposed as the sum of two individual contributions related to the discretization of the state and adjoint equations. We explore reliability and efficiency properties of the aforementioned error estimator. We illustrate the theory with numerical examples.

A posteriori error estimates for a bang-bang optimal control problem

TL;DR

The work develops a residual-type a posteriori error estimator for a PDE-constrained bang-bang optimal control problem analyzed without Tikhonov regularization, using variational discretization for the state and adjoint while keeping the control in the continuous admissible set. The estimator decomposes into state and adjoint components and is shown to be reliable and locally efficient on convex domains; a growth condition on the adjoint state yields a robust bound on the control and state errors in terms of the estimators, with a simplified form when the growth exponent equals 1. An adaptive finite element loop driven by the total estimator achieves optimal experimental convergence rates for all variables, and numerical experiments on convex and non-convex domains (including cases with unbounded forcing in the adjoint) validate the theory and demonstrate practical performance. The results provide a rigorous, practical framework for AFEM-based solution of bang-bang OCPs without regularization, including scenarios with unknown exact solutions and varying adjoint-growth behavior. The approach is well-suited for applications requiring accurate error control and efficient mesh refinement in PDE-constrained optimization with bang-bang controls.

Abstract

We propose and analyze a posteriori error estimates for a control-constrained optimal control problem with bang-bang solutions. We consider a solution strategy based on the variational approach, where the control variable is not discretized; no Tikhonov regularization is made. We design, for the proposed scheme, a residual-type a posteriori error estimator that can be decomposed as the sum of two individual contributions related to the discretization of the state and adjoint equations. We explore reliability and efficiency properties of the aforementioned error estimator. We illustrate the theory with numerical examples.
Paper Structure (14 sections, 4 theorems, 65 equations, 8 figures, 2 algorithms)

This paper contains 14 sections, 4 theorems, 65 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. The optimal control problem
  4. Weak formulation
  5. Finite element approximation
  6. A posteriori error estimates
  7. Reliability
  8. Auxiliary upper bounds
  9. Reliability estimates
  10. Efficiency
  11. Numerical examples
  12. Exact solution on convex domain
  13. Exact solution on non-convex domain
  14. Unknown solution on non-convex domain

Key Result

Theorem 3.1

Let $\bar{u}\in \mathbb{U}_{ad}$ be the unique solution to problem def:weak_ocp--eq:weak_st_eq with $\bar{y}$ and $\bar{p}$ being the corresponding state and adjoint state variables, respectively. Let $\bar{\mathfrak{u}}_{\ell}\in \mathbb{U}_{ad}$ be a solution to the semi-discrete problem with $\ba and The hidden constants are independent of the continuous and discrete optimal variables, the siz

Figures (8)

  • Figure 1: Experimental rates of convergence for individual contributions of the total error with uniform (1.A) and adaptive (1.B) refinement, convergence rates for individual contributions of the estimator $E$ (1.C), and effectivity index (1.D) with adaptive refinement for the problem from section \ref{['sec:ex_1']}.
  • Figure 2: Comparison of the continuous (red) and discrete switching sets on the adaptively refined meshes obtained after $5$ ((2.A) and (2.D)), $10$ ((2.B) and (2.E)), and $15$ ((2.C) and (2.F)) iterations for the problem from section \ref{['sec:ex_1']}.
  • Figure 3: Approximate control $\bar{\mathfrak{u}}_{\ell}$ obtained after $5$ (3.A), $10$ (3.B), and $15$ (3.C) iterations for the problem from section \ref{['sec:ex_1']}; in the red region the value is $1$ whereas in the blue region is $-1$.
  • Figure 4: Experimental rates of convergence for individual contributions of the total error with uniform (4.A) and adaptive (4.B) refinement, convergence rates for individual contributions of the estimator $E$ (4.C), and effectivity index (4.D) with adaptive refinement for the problem from section \ref{['sec:ex_2']}.
  • Figure 5: Comparison of the continuous (red) and discrete switching sets on the adaptively refined meshes obtained after $5$ ((5.A) and (5.D)), $10$ ((5.B) and (5.E)), and $15$ ((5.C) and (5.F)) iterations for the problem from section \ref{['sec:ex_2']}.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 3.1: reliability estimates
  • proof
  • Remark 3.2: case $\beta=1$
  • Lemma 3.3: local efficiency of ${\eta}_{st,2}$
  • proof
  • Lemma 3.4: local efficiency of ${\eta}_{adj,\infty}$
  • proof
  • Theorem 3.5: local efficiency