A distributed control problem for a fractional tumor growth model

TL;DR

This paper develops a framework for optimal control of a fractional tumor growth model, extending previous work by considering more general operators, and derives necessary conditions for optimality.

Contribution

It introduces a distributed control approach for a generalized fractional tumor growth system with new operator types, and establishes differentiability and optimality conditions.

Findings

Proved Fréchet differentiability of the control-to-state map.

Established existence of solutions to the adjoint system.

Derived first-order necessary optimality conditions.

Abstract

In this paper, the authors study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn-Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). 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 the recent work Adv. Math. Sci. Appl. 28 (2019), 343-375 (see arXiv:1906.10874), by the present authors. In the analysis, the Fr\'echet differentiability of the associated control-to-state operator is shown, by also establishing the existence of solutions to the associated adjoint system, and deriving the first-order necessary conditions of optimality for a cost functional of tracking type.

Full text

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A distributed control problem

for a fractional tumor growth model

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centerPierluigi Colli*(1)*

e-mail: pierluigi.colli@unipv.it

Gianni Gilardi*(1)*

e-mail: gianni.gilardi@unipv.it

Jürgen Sprekels*(2)*

e-mail: sprekels@wias-berlin.de

(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

(2) Department of Mathematics

Humboldt-Universität zu Berlin

Unter den Linden 6, 10099 Berlin, Germany

and

Weierstrass Institute for Applied Analysis and Stochastics

Mohrenstrasse 39, 10117 Berlin, Germany