| Abstract: |
| Phase-field models such as the Cahn--Hilliard system play a central role in the PDE-based modeling of phase separation phenomena in applied sciences. In this talk, we investigate an optimal control problem for a viscous Cahn--Hilliard system in which the chemical potential is subject to a hyperbolic relaxation.
We present recent analytical results for the associated state system, including well-posedness, regularity properties, and the Fr\`echet differentiability of the control-to-state mapping in suitable Banach spaces. Building on this framework, we study a sparse optimal control problem and derive first-order necessary conditions for local optimality.
Finally, we analyze the asymptotic behavior of the system as the relaxation parameter tends to zero, providing convergence results for the state variables, optimal controls, and adjoint system. |
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