Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth
arXiv:1903.00333 · doi:10.1051/cocv/2019059
Abstract
In this paper, we study 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 coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable, the chemical potential and the nutrient as well as a "no-friction" boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the phase field variable during the evolution and at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish 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.
arXiv admin note: substantial text overlap with arXiv:1811.07783
References in corpus (10)
- A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis
- Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
- Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
- Optimal distributed control of an extended model of tumor growth with logarithmic potential
- Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation
- Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach
- Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme
- Optimal control of a Vlasov-Poisson plasma by an external magnetic field
- Optimal control of a Vlasov-Poisson plasma by fixed magnetic field coils
- Pontryagin's maximum principle and second order optimality condition for optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions
Cited by in corpus (11)
- Viscoelastic Cahn--Hilliard models for tumour growth
- On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport
- Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials
- Optimal control of stochastic phase-field models related to tumor growth
- Penalisation of Long Treatment Time and Optimal Control of a Tumour Growth Model of Cahn-Hilliard
- Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential
- Optimal Control of the 3D Damped Navier-Stokes-Voigt Equations with Control Constraints
- Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities
- Vanishing parameter for an optimal control problem modeling tumor growth
- Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
- Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities