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Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

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Abstract

In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” by the same authors, general well-posedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous Cahn–Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators  A, B  having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in the quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.

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Acknowledgements

This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics “F. Casorati”, University of Pavia. In addition, PC and CG gratefully acknowledge some other financial support from the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, 27100, Pavia, Italy

    Pierluigi Colli & Gianni Gilardi

  2. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

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  1. Pierluigi Colli
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  2. Gianni Gilardi
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Correspondence to Pierluigi Colli.

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Colli, P., Gilardi, G. & Sprekels, J. Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System. Appl Math Optim 82, 551–589 (2020). https://doi.org/10.1007/s00245-018-9540-7

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