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Optimal control of a two-phase flow model with state constraints

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  • We investigate in this article the Pontryagin's maximum principle for a class of control problems associated with a two-phase flow model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity $v, $ coupled with a convective Allen-Cahn model for the order (phase) parameter $\phi. $ The optimal problems involve a state constraint similar to that considered in [18]. We derive the Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier-Stokes and the Allen-Cahn systems makes the analysis of the control problem more involved. In particular, the associated adjoint systems have less regularity than the one derived in [18].
    Mathematics Subject Classification: 35Q93, 49J21, 35A01, 35A02.

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