Distributed and boundary optimal control of the Allen–Cahn equation with regular potential and dynamic boundary conditions

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Fecha

2016

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Publicado en

International Journal of Control, 1366-5820, Vol 89, Nro 8, 2016, p 1523-1532

Publicado por

Taylor and Francis

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Abstract

This paper is concerned with an optimal control problem (P) (both distributed control as well as boundary control) for the nonlinear phase-field (Allen–Cahn) equation, involving a regular potential and dynamic boundary condition. A family of approximate optimal control problems (Pϵ) is introduced and results for the existence of an optimal control for problems (P) and (Pϵ) are proven. Furthermore, the convergence result of the optimal solution of problem (Pϵ) to the optimal solution of problem (P) is proved. Besides the existence of an optimal control in problem (Pϵ), necessary optimality conditions (Pontryagin's principle) as well as a conceptual gradient-type algorithm to approximate the optimal control, were established in the end.

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Keywords

Boundary value problems for nonlinear parabolic PDE, Dynamic boundary conditions, Pontryagin's maximum principle, Fractional steps method, Phase changes

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