Abstract
In this paper, we consider the problem for optimal control of the sixth-order convective Cahn-Hilliard type equation. The optimal control under boundary condition is given, the existence of an optimal solution to the equation is proved and the optimality system is established.
Highlights
In past decades, the optimal control of a distributed parameter system has received much more attention in academic field
A wide spectrum of problems in applications can be solved by the methods of optimal control such as chemical engineering and vehicle dynamics
The Cahn-Hilliard (CH) equation is a type of higher order nonlinear parabolic equation, it models many interesting phenomena in mathematical biology, fluid mechanics, phase transition, etc
Summary
The optimal control of a distributed parameter system has received much more attention in academic field. Many papers have already been published to study the control problems for nonlinear parabolic equations, for example, [ – ] and so on. By an extension of the method of matched asymptotic expansions that retains exponentially small terms, Korzec et al [ ] derived a new type of stationary solutions of the one-dimensional sixth-order Cahn-Hilliard equation. In Section , we consider the optimal control problem and prove the existence of an optimal solution. By analyzing the limiting behavior of sequences of a smooth function {un}, we can prove the existence of a weak solution to problem ( ). The equation of ( ) is an ordinary differential equation, and according to ODE theory, there exists a unique solution to problem ( ) in the interval [ , tn).
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