Optimal Control Problem for the Cahn–Hilliard/Allen–Cahn Equation with State Constraint

  • Xiaoli Zhang
  • Huilai Li
  • Changchun LiuEmail author


In this paper, we consider a distributed optimal control problem for the Cahn–Hilliard/Allen–Cahn equation with state-constraint. The objective is to force the coverage y to have some specified properties or achieve a certain goal. Since the cost functional is discontinuous, together with state constraint, we employ a new penalty functional by the approximation of the cost functional, in this case, we derive the necessary optimality conditions for the approximating optimal control problem. Finally, by considering the limits of the necessary optimality conditions we have obtained, we solve the optimal control problem and derive the necessary optimality conditions.


Cahn–Hilliard/Allen–Cahn equation State constraint Optimal control Optimality conditions Adjoint state system 

Mathematics Subject Classification

49J20 49K20 35K35 35K59 



The authors would like to express their deep thanks to the referee’s valuable suggestions for the revision and improvement of the manuscript.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsJilin UniversityChangchunChina

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