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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 0 and a prescribed stationary solution ψ, find a function f of u (feedback form) such that the corresponding solution to u t +γΔ 2 u=Δϕ(u)+f(u) subject ot the boundary conditions and initial condition will satisfy the following: lim t u(t)-ψ L 2 (Ω) =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 0 in L 2 and any given stationary solutions ψ, 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.
93D15Stabilization of systems by feedback
49K20Optimal control problems with PDE (optimality conditions)