scholarly journals Correction to: Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

2021 ◽  
Author(s):  
Pierluigi Colli ◽  
Andrea Signori ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
System Modelling ◽  
Singular Potentials ◽  
Field System ◽  
Phase Field System

scholarly journals Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth

2017 ◽  
Vol 10 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Elisabetta Rocca ◽  
Jürgen Sprekels ◽  
...  
Keyword(s):  
Tumor Growth ◽  
Error Estimates ◽  
Phase Field ◽  
System Modelling ◽  
Field System ◽  
Type Phase ◽  
Phase Field System ◽  
Asymptotic Analyses

scholarly journals Distributed optimal control of a nonstandard nonlocal phase field system

2016 ◽  
Vol 1 (3) ◽  
pp. 225-260 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Distributed Optimal Control ◽  
Field System ◽  
Phase Field System

scholarly journals Sliding Mode Control for a Phase Field System Related to Tumor Growth

2017 ◽  
Vol 79 (3) ◽  
pp. 647-670 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Gabriela Marinoschi ◽  
Elisabetta Rocca
Keyword(s):  
Sliding Mode Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Sliding Mode ◽  
Field System ◽  
Mode Control ◽  
Phase Field System

scholarly journals The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials

2010 ◽  
Vol 28 (1) ◽  
pp. 67-98 ◽  
Author(s):  
Maurizio Grasselli ◽  
◽  
Alain Miranville ◽  
Giulio Schimperna ◽  
◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Phase Field ◽  
Singular Potentials ◽  
Dynamic Boundary Conditions ◽  
Field System ◽  
Dynamic Boundary ◽  
Phase Field System

scholarly journals Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

2020 ◽  
Author(s):  
Jürgen Sprekels ◽  
Fredi Tröltzsch
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Optimality Conditions ◽  
Phase Field ◽  
Phase Field Model ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Sufficient Optimality Conditions ◽  
Phase Field System

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the $L^1$-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

scholarly journals Optimal control for a conserved phase field system with a possibly singular potential

2018 ◽  
Vol 7 (1) ◽  
pp. 95-116 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Gabriela Marinoschi ◽  
Elisabetta Rocca ◽  
...  
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Singular Potential ◽  
Field System ◽  
Phase Field System

scholarly journals Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

2017 ◽  
Vol 6 (1) ◽  
pp. 35-58 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Jürgen Sprekels ◽  
◽  
...  
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Distributed Optimal Control ◽  
Field System ◽  
Phase Field System
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