A Distributed Control Problem for a Fractional Tumor Growth Model

Pierluigi Colli, Gianni Gilardi and Jürgen Sprekels
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Pierluigi Colli: Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI—C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy
Gianni Gilardi: Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI—C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy
Jürgen Sprekels: Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

Mathematics, 2019, vol. 7, issue 9, 1-32

Abstract: In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

Keywords: fractional operators; Cahn–Hilliard systems; well-posedness; regularity; optimal control; necessary optimality conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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