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Distributed Optimal Control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq System for Nonisothermal Viscous Two-Phase Flows

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Abstract

We analyze a distributed optimal control problem where the state equation is governed by the coupling of the two-dimensional Cahn–Hilliard and Oberbeck–Boussinesq systems modelling incompressible viscous two-phase flows with convective heat transfer. Pointwise constraints are imposed on the controls that act as external sources in the fluid and convection–diffusion equations. The objective functional is of tracking-type that consists of a weighted energy of the difference between the state and a desired target. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, limited space–time regularity of the adjoint states arises due to convection and surface tension.

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Acknowledgements

The author would like to thank the anonymous referees for the helpful comments and suggestions that lead to the improvement of the paper.

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  1. Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600, Baguio, Philippines

    Gilbert Peralta

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  1. Gilbert Peralta
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This work was supported in part by the One U.P. Faculty Grant 2019-101374.

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Peralta, G. Distributed Optimal Control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq System for Nonisothermal Viscous Two-Phase Flows. Appl Math Optim 84 (Suppl 2), 1219–1279 (2021). https://doi.org/10.1007/s00245-021-09759-7

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