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Adaptive finite element method for an unregularized semilinear optimal control problem

Francisco Fuica, Nicolai Jork

Abstract

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational discretization approach. For this solution technique, we design an a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove, under suitable local growth conditions of optimal controls, reliability and efficiency properties of such error estimator. A simple adaptive strategy based on the devised estimator is designed and its performance is illustrated with numerical examples.

Adaptive finite element method for an unregularized semilinear optimal control problem

Abstract

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational discretization approach. For this solution technique, we design an a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove, under suitable local growth conditions of optimal controls, reliability and efficiency properties of such error estimator. A simple adaptive strategy based on the devised estimator is designed and its performance is illustrated with numerical examples.
Paper Structure (24 sections, 14 theorems, 103 equations, 10 figures, 1 algorithm)

This paper contains 24 sections, 14 theorems, 103 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. Results on the involved PDEs
  5. Linear elliptic PDE
  6. Semilinear elliptic PDE
  7. The optimal control problem
  8. Weak formulation
  9. The optimality conditions
  10. Some general remarks
  11. Finite element approximation
  12. Semidiscrete scheme
  13. A posteriori error analysis
  14. Reliability
  15. A posteriori error estimates: state equation
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $z\in L^r(\Omega)$ with $r>d/2$. Then, the linear equation eq:weak_st_lin has a unique solution $\varphi_{z}\in H^1_0(\Omega)\cap C(\bar{\Omega})$. In addition, it holds that where $C_r>0$ is independent of $a_0$ and $z$. Moreover, if $\Omega$ is convex, then $\varphi_z\in H^1_0(\Omega)\cap H^2(\Omega)$ and

Figures (10)

  • Figure 1: Experimental convergence rates for individual contributions of the total error with uniform (left) and adaptive (right) refinements for the problem from section \ref{['sec:ex_1']} with $f(\cdot,y)=\arctan(y)$.
  • Figure 2: Experimental convergence rates for individual contributions of the total error with uniform (left) and adaptive (right) refinements for the problem from section \ref{['sec:ex_1']} with $f(\cdot,y)=y^3$.
  • Figure 3: Approximate control $\bar{\mathfrak{u}}_{\ell}$ and comparison of the continuous (red) and discrete switching sets on the adaptively refined meshes obtained after $5$ (upper row) and $10$ (lower row) iterations for the problem from section \ref{['sec:ex_1']} with $f(\cdot,y)=\arctan(y)$; in the red region $\bar{\mathfrak{u}}_{\ell} = 1$ whereas in the blue region $\bar{\mathfrak{u}}_{\ell} = -1$.
  • Figure 4: Experimental convergence rates for individual contributions of the total error with uniform (left) and adaptive (right) refinements for the problem from section \ref{['sec:ex_2']}.
  • Figure 5: Experimental convergence rates for individual contributions of the estimator $\eta_{ocp}$ (left) and effectivity index (right) with adaptive refinement for the problem from section \ref{['sec:ex_2']}.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Lemma 2.1: well-posedness and stability
  • Lemma 2.2: stability in $L^{s}(\Omega)$
  • Theorem 2.4: well-posedness and regularity
  • Theorem 2.5: properties of $G_r$
  • proof
  • Proposition 3.1: properties of $J$
  • Theorem 3.2: first-order optimality conditions
  • Theorem 3.3: local optimality
  • Remark 3.4: more general problem
  • Remark 3.5: larger range for $\gamma$
  • ...and 15 more