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Strong Stationarity Conditions for the Optimal Control of a Cahn–Hilliard–Navier–Stokes System

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Abstract

This paper is concerned with the distributed optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier–Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.

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Funding

Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SPP-1962), DFG-Research Center MATHEON (Grant No. C-SE5), and Einstein Stiftung Berlin (Grant No. ECMATH: D-OT1).

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Michael Hintermüller & Tobias Keil

  2. Institute for Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller & Tobias Keil

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  1. Michael Hintermüller
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Correspondence to Michael Hintermüller.

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This research was supported by the German Research Foundation DFG through the SPP 1506 and the SPP1962 and by the Research Center MATHEON through project C-SE5 and D-OT1 funded by the Einstein Center for Mathematics Berlin.

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Hintermüller, M., Keil, T. Strong Stationarity Conditions for the Optimal Control of a Cahn–Hilliard–Navier–Stokes System. Appl Math Optim 89, 12 (2024). https://doi.org/10.1007/s00245-023-10063-9

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