Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems

• Autorzy: Colli Pierluigi , Gilardi Gianni , Sprekels Jurgen
• Języki publikacji: EN

Abstrakty

EN

In this paper, we give an overview of results for Cahn–Hilliard systems involving fractional operators that have recently been established by the authors of this note. We address problems concerning existence, uniqueness, and regularity of the solutions to the system equations, and we study optimal control problems for the systems. The well-posedness results are valid for a wide class of fractional operators of spectral type and for the typical double-well nonlinearities appearing in the Cahn–Hilliard system equations, namely the classical differentiable, the logarithmic, and the nondifferentiable double obstacle potentials. While this also applies to the existence of optimal controls in the related optimal control problems, the establishment of first-order necessary optimality conditions requires imposing much stronger assumptions on the admissible class of fractional operators. One main reason for this is the necessity of deriving suitable differentiability properties for the associated control-to-state mapping. Nevertheless, it turns out that also in the singular case of logarithmic potentials, the first-order necessary optimality conditions can be established under suitable assumptions, and a “deep quench” approximation, based on the results derived for logarithmic nonlinearities makes even the case of double obstacle potentials accessible.

Słowa kluczowe

EN

fractional operators; Cahn-Hilliard system; optimal control; necessary optimality conditions

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 1
• Strony: 153--197
• Opis fizyczny: Bibliogr. 65 poz., rys., tab.

Twórcy

autor

Colli Pierluigi

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

autor

Gilardi Gianni

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

autor

Sprekels Jurgen

• Department of Mathematics Humboldt-Universit¨at zu Berlin Unter den Linden 6, 10099 Berlin, Germany
• Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39, 10117 Berlin, Germany

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Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).