22 research outputs found
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Analysis of a compressible Stokes-flow with degenerating and singular viscosity
In this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains
A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media
In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous
Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori
estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase
field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional
penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces
acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of
minimizers and study a family of "local'' approximations via adapted cost functionals
Analysis of a compressible Stokes-flow with degenerating and singular viscosity
In this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains
Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D
Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749--2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system
Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions
In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem
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A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media
In this work we investigate a phase field model for damage processes in
two-dimensional viscoelastic media with nonhomogeneous Neumann data
describing external boundary forces. In the first part we establish
global-in-time existence, uniqueness, a priori estimates and continuous
dependence of strong solutions on the data. The main difficulty is caused by
the irreversibility as well as boundedness of the phase field variable which
results in a doubly constrained PDE system. In the last part we consider an
optimal control problem where a cost functional penalizes maximal deviations
from prescribed damage profiles. The goal is to minimize the cost functional
with respect to exterior forces acting on the boundary which play the role of
the control variable in the considered model . To this end, we prove
existence of minimizers and study a family of "local'' approximations via
adapted cost functionals
A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles
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
Chance constraints in PDE constrained optimization
Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints
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Singular limit of Allen-Cahn equation with constraints and its Lagrange multiplier
We consider the Allen-Cahn equation with constraint. Our constraint is
the subdifferential of the indicator function on the closed interval, which
is the multivalued function. In this paper we give the characterization of
the Lagrange multiplier to our equation. Moreover, we consider the singular
limit of our system and clarify the limit of the solution and the Lagrange
multiplier to our problem
Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier
We consider the Allen-Cahn equation with constraint. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier to our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our problem