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Optimal control of a new mechanochemical model with state constraint
Author(s) -
Liu Changchun,
Zhang Xiaoli
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7350
Subject(s) - optimal control , mathematics , constraint (computer aided design) , state (computer science) , mathematical optimization , control (management) , computer science , algorithm , geometry , artificial intelligence
In this paper, a state‐constrained optimal control problem is considered for the new mechanochemical model in biological patterns modelling the skin coating of some vertebrate marine animals. The model consists of the time‐dependent Ginzburg–Landau equation for the concentration difference of at least two pigments u coupled with the Swift–Hohenberg equation for the difference of dermal cellular densities of at least two types of cells. The objective is to force the state to be as close as possible to a desired state. Considering that the cost functional is discontinuous, which along with state constraint, we apply a new penalty functional to approximate the cost functional, in this case, so we first prove the existence of optimal control. Then we derive the necessary optimality conditions for the approximating optimal control problem. Finally, the necessary optimality conditions is obtained by considering the limits of the results for the approximating optimal control problem.