| Abstract: |
| We consider a Cahn-Hilliard-Darcy system with mass sources, equipped
with an impermeability condition for the (volume) averaged velocity as
well as homogeneous Neumann boundary conditions for the phase field
and the chemical potential. The source term in the convective
Cahn-Hilliard equation contains a control R that can be thought, for
instance, as a drug or a nutrient in applications to solid tumor
growth evolution. We present some recent results obtained in
collaboration with M. Abatangelo, M. Grasselli, and H. Wu on a
distributed optimal control problem in the two dimensional setting
with a cost functional of tracking-type. These results have been
achieved In the physically relevant case, that is, assuming unmatched
viscosities for the binary fluid mixtures and considering a
Flory-Huggins type potential. In particular, we show that a
second-order sufficient condition for the strict local optimality can
also be proven. |
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