A Boundary Control Problem for the Equation and Dynamic Boundary Condition of Cahn–Hilliard Type
A dynamic boundary condition is a type of partial differential equation that describes the dynamics of a system on the boundary. Combining with the heat equation in a smooth-bounded domain, the characteristic structure of “total mass conservation” appears, namely, the volume in the bulk plus the volume on the boundary is conserved. Based on this interesting structure, an equation and dynamic boundary condition of Cahn–Hilliard type was introduced by Goldstein–Miranville–Schimperna. In this paper, based on the previous work of Colli–Gilardi–Sprekels, a boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type is considered. The optimal boundary control that realizes the minimal cost under a control constraint is determined, and a necessary optimality condition is obtained.
KeywordsBoundary control Cahn–Hilliard system Dynamic boundary condition
AMS (MOS) Subject Classification49J20 35K51 34G25
This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. The authors are grateful to the anonymous referee for reviewing the original manuscript and for valuable comments that helped to clarify and refine this paper. This work is supported by JSPS KAKENHI Grants-in-Aid for Scientific Research(C), Grant Number 26400164 for TF and 26400179 for NY.
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