Vanishing parameter for an optimal control problem modeling tumor growth

Article type: Research Article

Authors: Signori, Andrea^{; *}

Affiliations: Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca via Cozzi 55, 20125 Milano, Italy. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn–Hilliard equation for the phase variable (relative concentration of the tumor) coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity at which it is possible to dispense medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order necessary optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero.

Keywords: Asymptotic analysis, distributed optimal control, phase field model, tumor growth, cancer treatment, evolution equations, Cahn–Hilliard equation, optimal control, adjoint system, necessary optimality conditions

DOI: 10.3233/ASY-191546

Journal: Asymptotic Analysis, vol. 117, no. 1-2, pp. 43-66, 2020