WIAS Preprint No. 2228, (2016)

Optimal distributed control of a diffuse interface model of tumor growth


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Rocca, Elisabetta
    ORCID: 0000-0002-9930-907X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 35K61 49J20 49K20 92C50


  • Distributed optimal control, first-order necessary optimality conditions, tumor growth, reaction-diffusion equations, Cahn--Hilliard equation




In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn--Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

Appeared in

  • Nonlinearity, 30 (2017), pp. 2518--2546.

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