Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth
Abstract
:1. Introduction
2. Functional Setting, Assumptions and Previous Results
- The potential function is such that where and satisfying for all and . In addition, we assume that for all and
- The proliferation function satisfies either one of the following properties for all
3. Parameters Identification and Optimal Problem
3.1. Study of the Linearized-State System
3.2. Fréchet Differentiability of the Control to State Map
3.3. The Adjoint System
3.4. Necessary Optimality Condition
4. Numerical Illustration
Algorithm 1 Gauss-Newton scheme |
procedureGauss–Newton(, ) |
, , |
while ( and and ( and ) and ( and ) do |
for do |
Solve the problem (22) |
end for |
Find such that |
with |
if and and then |
Stop |
end if |
end while |
return |
end procedure |
4.1. Validation Test
4.2. Tumor Growth Computation
4.2.1. Two-Dimensional (2D) Case
4.2.2. Three-Dimensional (3D) Case
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Kadiri, M.; Louaked, M.; Trabelsi, S. Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth. Mathematics 2023, 11, 1607. https://doi.org/10.3390/math11071607
Kadiri M, Louaked M, Trabelsi S. Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth. Mathematics. 2023; 11(7):1607. https://doi.org/10.3390/math11071607
Chicago/Turabian StyleKadiri, Mostafa, Mohammed Louaked, and Saber Trabelsi. 2023. "Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth" Mathematics 11, no. 7: 1607. https://doi.org/10.3390/math11071607
APA StyleKadiri, M., Louaked, M., & Trabelsi, S. (2023). Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth. Mathematics, 11(7), 1607. https://doi.org/10.3390/math11071607