Optimal distributed control of a diffuse interface model of tumor growth *

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 Hawkins–Daruud et al (2011 Int. J. Numer. Math. Biomed. Eng . 28 3–24). The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side of the equation for σ 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.