M. Hassan Farshbaf-Shaker

Weierstrass Institute for Applied Analysis and Stochastics

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Pierluigi Colli

University of Pavia

Jürgen Sprekels

Weierstrass Institute for Applied Analysis and Stochastics

Luise Blank

University of Regensburg

Christoph Rupprecht

Ehime University

Gianni Gilardi

University of Pavia

Harald Garcke

University of Regensburg

Christian Heinemann

Weierstrass Institute for Applied Analysis and Stochastics

Claudia Hecht

University of Regensburg

Noriaki Yamazaki

Kanagawa University

Vanessa Styles

University of Sussex

Cooperative Institutions

Weierstrass Institute for Applied Analysis and Stochastics

Humboldt-Universität zu Berlin

University of Regensburg

University of Pavia

Politecnico di Milano

Istituto di Matematica Applicata e Tecnologie Informatiche

University of Rome Tor Vergata

Technische Universität Berlin

Technion – Israel Institute of Technology

Humboldt State University

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Optimal Neumann Boundary Control of a Vibrating String with Uncertain Initial Data and Probabilistic Terminal Constraints

SIAM Journal on Control and Optimization (2020)

M. Hassan Farshbaf-Shaker Martin Gugat Holger Heitsch René Henrion

In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for a control that steers the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. Numerical examples with optimal Neumann control of the wave equation are presented.

10.1137/19m1269944

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Citations (17)

Optimal Control of Doubly Nonlinear Evolution Equations Governed by Subdifferentials Without Uniqueness of Solutions

IFIP advances in information and communication technology (2016)

M. Hassan Farshbaf-Shaker Noriaki Yamazaki

10.1007/978-3-319-55795-3_24

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Citations (1)

Optimal Boundary Control of a Viscous Cahn--Hilliard System with Dynamic Boundary Condition and Double Obstacle Potentials

SIAM Journal on Control and Optimization (2015)

Pierluigi Colli M. Hassan Farshbaf-Shaker Gianni Gilardi Jürgen Sprekels

In this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, and Sprekels [Appl. Math. Optim., 71 (2015), pp. 1--24] to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, and Sprekels [Appl. Math. Optim., Online First DOI:10.1007/s00245-015-9299-z, 2015] for the case of (differentiable) logarithmic potentials and perform a so-called deep quench limit. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

Operator (biology)

Obstacle problem

10.1137/140984749

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Citations (70)

A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary condition and double obstacle potentials

arXiv (Cornell University) (2013)

Pierluigi Colli M. Hassan Farshbaf-Shaker Jürgen Sprekels

In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

Allen–Cahn equation

10.48550/arxiv.1308.5617

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Citations (3)

Relating phase field and sharp interface approaches to structural topology optimization

ESAIM Control Optimisation and Calculus of Variations (2014)

Luise Blank Harald Garcke M. Hassan Farshbaf-Shaker Vanessa Styles

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.

Topology optimization

Void (composites)

Shape Optimization

10.1051/cocv/2014006

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Citations (50)

Optimal control of Allen-Cahn systems

arXiv (Cornell University) (2013)

Luise Blank M. Hassan Farshbaf-Shaker Claudia Hecht Josef Michl Christoph Rupprecht

Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an obstacle potential first numerical results are presented.

Obstacle problem

Source

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Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes

Discrete and Continuous Dynamical Systems - S (2010)

M. Hassan Farshbaf-Shaker Harald Garcke

In this paper we derive thermodynamically consistent higher order phase field models for the dynamics of biomembranes in incompressible viscous fluids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [3] and Jamet and Misbah [11]. In particular we derive a stress tensor involving higher order derivatives of the phase field and generalize the classical Korteweg capillarity tensor.

Conservation law

Phase-Field Models

Viscous stress tensor

10.3934/dcdss.2011.4.371

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Citations (9)

Optimal Control for Shape Memory Alloys of the One-Dimensional Frémond Model

Numerical Functional Analysis and Optimization (2020)

Pierluigi Colli M. Hassan Farshbaf-Shaker Ken Shirakawa Noriaki Yamazaki

In this article, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for approximating systems.

10.1080/01630563.2020.1774892

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Citations (1)

Optimal control of Allen-Cahn systems

arXiv (Cornell University) (2013)

Luise Blank M. Hassan Farshbaf-Shaker Claudia Hecht Josef Michl Christoph Rupprecht

Obstacle problem

10.48550/arxiv.1312.2351

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Citations (0)

Multi-material phase field approach to structural topology optimization

arXiv (Cornell University) (2013)

Luise Blank M. Hassan Farshbaf-Shaker Harald Garcke Christoph Rupprecht Vanessa Styles

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an $H^1$-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.

Topology optimization

Shape Optimization

Projection method

10.48550/arxiv.1312.2356

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