Volume 26, 2020
|Number of page(s)||38|
|Published online||30 September 2020|
Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth
Department of Mathematics, University of Regensburg,
* Corresponding author: firstname.lastname@example.org
Accepted: 23 September 2019
We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.
Mathematics Subject Classification: 35K61 / 76D07 / 49J20 / 92C50
Key words: Optimal control with PDEs / calculus of variations / tumor growth / Cahn–Hilliard equation / Brinkman equation / first-order necessary optimality conditions / second-order sufficient optimality conditions / uniqueness of globally optimal solutions
© EDP Sciences, SMAI 2020
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