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Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability and optimal control

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg, Gerd Wachsmuth

TL;DR

This work analyzes obstacle-type quasi-variational inequalities by focusing on the minimal and maximal solution maps $\mathsf{m}$ and $\mathsf{M}$ and their penalised counterparts. It proves local Lipschitz stability and directional (Hadamard) differentiability for these extremal maps, and develops a Moreau–Yosida-type penalisation that yields constructive, PDE-based approximations to the extremal solutions. The penalised maps are shown to converge to the QVI maps as $\rho\searrow 0$, enabling a rigorous derivation of Bouligand and C-stationarity conditions for related optimal-control problems. The results include a detailed characterisation of derivatives via QVI-type linearizations, constructive iteration schemes, and applications to obstacle-map and thermoforming models, providing both theoretical guarantees and computational pathways for non-unique QVIs.

Abstract

Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau--Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.

Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability and optimal control

TL;DR

This work analyzes obstacle-type quasi-variational inequalities by focusing on the minimal and maximal solution maps and and their penalised counterparts. It proves local Lipschitz stability and directional (Hadamard) differentiability for these extremal maps, and develops a Moreau–Yosida-type penalisation that yields constructive, PDE-based approximations to the extremal solutions. The penalised maps are shown to converge to the QVI maps as , enabling a rigorous derivation of Bouligand and C-stationarity conditions for related optimal-control problems. The results include a detailed characterisation of derivatives via QVI-type linearizations, constructive iteration schemes, and applications to obstacle-map and thermoforming models, providing both theoretical guarantees and computational pathways for non-unique QVIs.

Abstract

Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau--Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.
Paper Structure (27 sections, 43 theorems, 236 equations)

This paper contains 27 sections, 43 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Main results
  4. On directional differentiability
  5. On the penalised problem
  6. On optimal control
  7. Examples
  8. Obstacle map as solution map of PDE
  9. An application in thermoforming
  10. Properties of the penalised problem
  11. Results on a semilinear elliptic PDE
  12. Order properties
  13. Minimal and maximal solutions of PDEs
  14. Convergence to the QVIs
  15. Properties with respect to varying $\rho$
  16. ...and 12 more sections

Key Result

Proposition 1.2

Under ass:forZSubAndSuperSolutions, there exist a minimal solution $\mathsf{m}(f)$ and maximal solution $\mathsf{M}(f)$ to eq:QVIIntro on the interval $[\underline u, \overline u] := \set{v \in V \nonscript\;\delimsize|\nonscript\;\mathopen{} \underline u \le v \le \overline u \text{ a.e.\ in } \Ome

Theorems & Definitions (95)

  • Proposition 1.2
  • proof
  • Remark 2.2
  • Theorem 2.3: Local Lipschitz continuity of $\mathsf{Z}$
  • Theorem 2.4: Hadamard differentiability of $\mathsf{Z}$
  • Remark 2.5: Directional differentiability of $\mathsf{Z}$
  • Example 2.6: The radial cone $\mathcal{R}_W(f)$
  • Theorem 2.8: Convergence of $\mathsf{Z}_\rho(g)$ to $\mathsf{Z}(f)$
  • Corollary 2.9: Convergence of $\mathsf{Z}_\rho(f)$ to $\mathsf{Z}(f)$
  • Remark 2.10
  • ...and 85 more