Volume 29, 2023
|Number of page(s)||43|
|Published online||01 May 2023|
Optimal Borel Measure-Valued Controls to the Viscous Cahn–Hilliard–Oberbeck–Boussinesq Phase-Field System on Two-Dimensional Bounded Domains
Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio, Philippines
* Corresponding author: email@example.com
Accepted: 31 March 2023
We consider an optimal control problem for the two-dimensional viscous Cahn-Hilliard-Ober beck-Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equations governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.
Mathematics Subject Classification: 35B65 / 35Q93 / 49K20 / 76D55
Key words: Cahn-Hilliard equation / Oberbeck-Boussinesq system / binary fluids / interpolation spaces / Borel measures / sparsity / optimality conditions / Lagrangian
© The authors. Published by EDP Sciences, SMAI 2023
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