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Author: Krumbiegel, Klaus × Type: Report ×

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Now showing 1 - 6 of 6
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    A priori error analysis for state constrained boundary control problems : Part II: Full discretization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krumbiegel, Klaus; Meyer, Christian; Rösch, Arnd
    This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations.
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    Regularization error estimates for semilinear elliptic optimal control problems with pointwise state and control constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Krumbiegel, Klaus; Neitzel, Ira; Rösch, Arnd
    In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. A sufficient second order optimality condition and uniqueness of the dual variables are assumed for that problem. Sufficient second order optimality conditions are shown for regularized problems with small regularization parameter. Moreover, error estimates with respect to the regularization parameter are derived
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    A priori error analysis for state constrained boundary control problems : Part I: Control discretization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krumbiegel, Klaus; Meyer, Christian; Rösch, Arnd
    This is the first of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [20] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems
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    Sufficient optimality conditions for the Moreau-Yosida-type regularization concept applied to semilinear elliptic optimal control problems with pointwise state constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Krumbiegel, Klaus; Neitzel, Ira; Rösch, Arnd
    We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Moreover, we compare both regularization approaches within a numerical example
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    Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Krumbiegel, Klaus; Rehberg, Joachim
    In this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three. Moreover, we consider pointwise constraints on the control and on the state. We formulate first order necessary and second order sufficient optimality conditions. We make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations.
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    Boundary coefficient control : a maximal parabolic regularity approach
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Hömberg, Dietmar; Krumbiegel, Klaus; Rehberg, Joachim
    We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.