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

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

An induction theorem and nonlinear regularity models

A general nonlinear regularity model for a set-valued mapping $F:X\times R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping $F:X\rightrightarrows Y$.Comment: 28 page

Optimal distributed control of a generalized fractional Cahn-Hilliard system

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors, general well-posedness results have been established for a a class of evolutionary systems of two equations having the structure of a viscous Cahn-Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in arXiv:1804.11290 by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal control, necessary optimality condition

Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in 2D2D

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen--Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier--Stokes equations for the fluid velocity coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.Comment: 41 pages, 24 figures; moderate revision after the first round of review, Appendix A re-written completel

A formulation of the fractional Noether-type theorem for multidimensional Lagrangians

This paper presents the Euler-Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses well-known notion of the Riemann-Liouville fractional derivative.Comment: Submitted 26-SEP-2011; accepted 3-MAR-2012; for publication in Applied Mathematics Letter

Fractional Calculus of Variations for Double Integrals

We consider fractional isoperimetric problems of calculus of variations with double integrals via the recent modified Riemann-Liouville approach. A necessary optimality condition of Euler-Lagrange type, in the form of a multitime fractional PDE, is proved, as well as a sufficient condition and fractional natural boundary conditions.Comment: Submitted 07-Sept-2010; revised 25-Nov-2010; accepted 07-Feb-2011; for publication in Balkan Journal of Geometers and Its Applications (BJGA