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Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations

Alberto Domínguez Corella, Nicolai Jork, Vladimir Veliov

TL;DR

This work studies stability of affine optimal control problems constrained by semilinear elliptic PDEs through strong Hölder subregularity of the first-order optimality system. It develops differentiability results for the state, adjoint, and switching mappings, and derives a robust framework that tolerates nonlinear perturbations with weaker structural assumptions. A key contribution is establishing strong Hölder subregularity of the optimality mapping with exponent $1/k^*$, yielding explicit Hölder error bounds and convergence rates. The results are then applied to nonlinear perturbations and Tikhonov regularization, providing quantitative error estimates and convergence rates that enhance error analysis and numerical stability in PDE-constrained control. The analysis is complemented by equivalent subregularity conditions and practical criteria (e.g., extended cones and measure-type conditions) to verify the assumptions in concrete settings.

Abstract

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.

Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations

TL;DR

This work studies stability of affine optimal control problems constrained by semilinear elliptic PDEs through strong Hölder subregularity of the first-order optimality system. It develops differentiability results for the state, adjoint, and switching mappings, and derives a robust framework that tolerates nonlinear perturbations with weaker structural assumptions. A key contribution is establishing strong Hölder subregularity of the optimality mapping with exponent , yielding explicit Hölder error bounds and convergence rates. The results are then applied to nonlinear perturbations and Tikhonov regularization, providing quantitative error estimates and convergence rates that enhance error analysis and numerical stability in PDE-constrained control. The analysis is complemented by equivalent subregularity conditions and practical criteria (e.g., extended cones and measure-type conditions) to verify the assumptions in concrete settings.

Abstract

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.
Paper Structure (16 sections, 37 theorems, 172 equations)

This paper contains 16 sections, 37 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The elliptic operator
  4. The control model
  5. Differentiability of the mappings involved
  6. The state and adjoint mappings
  7. The switching mapping
  8. Stability
  9. The main assumption
  10. Some existence and stability results
  11. Strong metric subregularity
  12. Nonlinear Perturbations
  13. The perturbations
  14. The stability result
  15. An application: Tikhonov regularization
  16. ...and 1 more sections

Key Result

Lemma 2.1

The set $D(\mathcal{L})$ is a linear subspace of $H^1(\Omega)\cap L^\infty(\Omega)$. Moreover, the operator $\mathcal{L}:D(\mathcal{L})\to L^2(\Omega)$ is a well defined linear mapping.

Theorems & Definitions (70)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 60 more