The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
  control, double obstacles, necessary optimality condition

Colli, Pierluigi

Gilardi, Gianni

Sprekels, Jürgen

English

arXiv

In the recent paper ''Well-posedness and regularity for a generalized fractional Cahn--Hilliard system'', the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn--Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn--Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper ''Optimal distributed control of a generalized fractional Cahn--Hilliard system'' (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic double-well potentials could be admitted. Results concerning existence of optimizers and first-order necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated control-to-state operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called ''deep quench'' method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper ''A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles'' (Appl. Math. Optim. 71 (2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of Cahn--Hilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

Deep quench approximation and optimal control of general Cahn--Hilliard systems with fractional  operators and double obstacle potentials

In the recent paper Well-posedness and regularity for a generalized
fractional CahnHilliard system, the same authors derived general
well-posedness and regularity results for a rather general system of
evolutionary operator equations having the structure of a CahnHilliard
system. The operators appearing in the system equations were fractional
versions in the spectral sense of general linear operators A and B having
compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process
modeled by the CahnHilliard system could be of a very general type that
includes standard physically meaningful cases such as polynomial,
logarithmic, and double obstacle nonlinearities. In the subsequent paper
Optimal distributed control of a generalized fractional CahnHilliard system
(Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the
same authors, an analysis of distributed optimal control problems was
performed for such evolutionary systems, where only the differentiable case
of certain polynomial and logarithmic double-well potentials could be
admitted. Results concerning existence of optimizers and first-order
necessary optimality conditions were derived, where more restrictive
conditions on the operators A and B had to be assumed in order to be able to
show differentiability properties for the associated control-to-state
operator. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called deep quench method. This
technique, in which the nondifferentiable double obstacle nonlinearity is
approximated by differentiable logarithmic nonlinearities, was first
developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper A
deep quench approach to the optimal control of an AllenCahn equation with
dynamic boundary conditions and double obstacles (Appl. Math. Optim. 71
(2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal
control problems with double obstacle potentials in the framework of systems
of CahnHilliard type. We first give a general convergence analysis of the
deep quench approximation that includes an error estimate and then
demonstrate that its use leads in the double obstacle case to appropriate
first-order necessary optimality conditions in terms of a variational
inequality and the associated adjoint state system

Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials

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