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Stability and Sensitivity Analysis for Quasi-Variational Inequalities

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Non-Smooth and Complementarity-Based Distributed Parameter Systems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 172))

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Abstract

We discuss various aspects of quasi-variational inequalities (QVIs) related to their sensitivity analysis and optimal control. Starting with the necessary functional framework and existence results for elliptic QVIs of obstacle type, we study stability of the solution map taking the source term onto the set of solutions: we show that certain realisations of the map have appropriate continuity properties. We then focus on showing that a notion of directional derivative exists for QVIs and we characterise this derivative as a monotone limit of directional derivatives associated to particular variational inequalities. The differentiability theory is illustrated with a novel application in thermoforming. Using the stability results, we discuss control problems with QVI constraints and prove existence of optimal controls.

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Notes

  1. 1.

    In particular, if A is linear, this is equivalent to 〈Ay−, y+〉≤ 0 for all y ∈ V , and we have the availability of maximum principles for A.

  2. 2.

    In fact, (A1) can be weakened significantly by requiring Hadamard differentiability of Φ only around the point y, i.e., locally, as in assumption (L1).

  3. 3.

    In this case, solutions of the QVI (3.6) are unique [23].

  4. 4.

    Zero Dirichlet conditions arise from clamping the membrane at its ends.

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Author information

Authors and Affiliations

  1. Weierstrass Institute, Berlin, Germany

    Amal Alphonse

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Michael Hintermüller

  3. Department of Mathematical Sciences and the Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, Fairfax, VA, USA

    Carlos N. Rautenberg

Authors
  1. Amal Alphonse
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Correspondence to Michael Hintermüller .

Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Michael Hintermüller

  2. Institute for Applied Mathematics, University of Heidelberg, Heidelberg, Germany

    Roland Herzog

  3. Lehrstuhl für Mathematik VII, Julius-Maximilians-Universität Würzburg, Würzburg, Germany

    Christian Kanzow

  4. Department of Mathematics, Technical University of Munich, Garching b. München, Germany

    Michael Ulbrich

  5. Fachbereich Mathematik, Technische Universitaet Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Alphonse, A., Hintermüller, M., Rautenberg, C.N. (2022). Stability and Sensitivity Analysis for Quasi-Variational Inequalities. In: Hintermüller, M., Herzog, R., Kanzow, C., Ulbrich, M., Ulbrich, S. (eds) Non-Smooth and Complementarity-Based Distributed Parameter Systems. International Series of Numerical Mathematics, vol 172. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-79393-7_8

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