A boundary control problem for a possibly singular phase field system with dynamic boundary conditions

This paper deals with an optimal control problem related to a phase field system of Caginalp type with a dynamic boundary condition for the temperature. The control placed in the dynamic boundary condition acts on a part of the boundary. The analysis carried out in this paper proves the existence of an optimal control for a general class of potentials, possibly singular. The study includes potentials for which the derivatives may not exist, these being replaced by well-defined subdifferentials. Under some stronger assumptions on the structure parameters and on the potentials (namely for the regular and the logarithmic case having single-valued derivatives), the first order necessary optimality conditions are derived and expressed in terms of the boundary trace of the first adjoint variable.Comment: Key words: phase field system, phase transition, singular potentials, optimal control, optimality conditions, adjoint state system, dynamic boundary conditions. arXiv admin note: text overlap with arXiv:1410.671

The H∞H^{\infty}-control problem for parabolic systems with singular Hardy potentials

We solve the $H^{\infty}$-control problem with state feedback for infinite dimensional boundary control systems of parabolic type with distributed disturbances and apply the results to equations with Hardy potentials with the singularity inside or on the boundary, in the cases of a distributed control and of a boundary control.Comment: 37 pages, 0 figure

Feedback stabilization of the Cahn-Hilliard type system for phase separation

This article is concerned with the internal feedback stabilization of the phase field system of Cahn-Hilliard type, modeling the phase separation in a binary mixture. Under suitable assumptions on an arbitrarily fixed stationary solution, we construct via spectral separation arguments a feedback controller having the support in an arbitrary open subset of the space domain, such that the closed loop nonlinear system exponentially reach the prescribed stationary solution. This feedback controller has a finite dimensional structure in the state space of solutions. In particular, every constant stationary solution is admissible.Comment: 43 pages. Keywords: Cahn-Hilliard system, Feedback control, Closed loop system, Stabilizatio

A Trotter–Kato type theorem in the weak topology and an application to a singular perturbed problem

AbstractIn this paper we prove a result of the Trotter–Kato type in the weak topology. Let {Aε}ε>0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by Jλε the resolvent of Aε. Under certain conditions, the result states that if for any x∈X and k=1,2,…, the sequence (Jλε)kx converges weakly to (Jλ)kx as ε→0, where Jλ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by −Aε tends weakly to the semigroup generated by −A, uniformly with respect to t on compact intervals. The result is different from other results of the same type (see e.g., Yosida (1980) [9, p. 269]) and gives an answer to an open problem put in Eisner and Serény (2010) [3]. It is finally applied to compare the asymptotic behavior of a singular perturbation problem associated to a first order hyperbolic problem with diffusion