(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, Jürgen
In this paper, a distributed optimal control problem is studied for a
diffuse interface model of tumor growth which was proposed by HawkinsDaruud
et al. in [25]. The model consists of a CahnHilliard equation for the tumor
cell fraction 'coupled to a reaction-diffusion equation for a function phi
representing the nutrientrich extracellular water volume fraction. The
distributed control u monitors as a right-hand side the equation for sigma
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.
