Abstract
In this paper, we shall study the optimal control of the initial-boundary value problem of a higher-order nonlinear parabolic equation describing crystal surface growth. The existence and uniqueness of weak solutions to the problem are given. According to the variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of the solution is related to the control item and initial value in the special Hilbert space. The optimal control of the problem is given, the existence of optimal solution is proved and the optimality system is established.
Highlights
The field of optimal control was born in the 1950s with the discovery of the maximum principle as a result of a competition in military affairs in the early days of the cold war
In the papers wrote by Ryu and Yagi [13, 14], the optimal control problems of Keller–Segel equations and adsorbate-induced phase transition model were
It is well known that the optimality conditions for w is given by the variational inequality
Summary
The field of optimal control was born in the 1950s with the discovery of the maximum principle as a result of a competition in military affairs in the early days of the cold war. Many papers have already been published to study the control problems of nonlinear parabolic equations. In [17], Yong and Zheng considered the feedback stabilization and optimal control of the Cahn–Hilliard equation in a bounded domain with smooth boundary. It was Rost and Krug [11] who studied the unstable epitaxy on singular surfaces using equation (3) with a prescribed slope dependent surface current In their paper, they derived scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed. In [18], Zhao and Cao studied the optimal control problem for equation (3) in 1D case. We consider the optimal control problem for two-dimensional equation (3) together with the initial and boundary conditions.
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