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Feedback stabilization and optimal control for the Cahn-Hilliard equation. (English) Zbl 0765.93067
This paper is concerned with the control problems for the Cahn-Hilliard equation which arises from the study of phase transitions. 1. Feedback stabilization problem: For a given initial datum $u\sb 0$ and a prescribed stationary solution $\psi$, find a function $f$ of $u$ (feedback form) such that the corresponding solution to $u\sb t+\gamma\Delta\sp 2 u=\Delta\varphi(u)+f(u)$ subject ot the boundary conditions and initial condition will satisfy the following: $\lim\sb{t\to\infty}\Vert u(t)-\psi\Vert\sb{L\sp 2(\Omega)}=0$. 2. Optimal control problem. Especially, we are interested in the characterization of the optimal control if it exists. The results obtained in this paper show that for any initial data $u\sb 0$ in $L\sp 2$ and any given stationary solutions $\psi$, there exists a explicit feedback form $f$ such the convergence indicated in above is exponentially fast. For the optimal control problem, the weak bang-bang principle is proved.
Reviewer: J.Yong (Shanghai)

MSC:
93D15Stabilization of systems by feedback
49K20Optimal control problems with PDE (optimality conditions)
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References:
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