2017, Gubisch, Martin, Neitzel, Ira, Volkwein, Stefan

In this work a-posteriori error estimates for linear-quadratic optimal control problems governed by parabolic equations are considered. Different error estimation techniques for finite element discretizations and model-order reduction are combined to validate suboptimal control solutions from low-order models which are constructed by a Galerkin discretization and the application of proper orthogonal decomposition. The theoretical findings are used to design an updating algorithm for the reduced-order models; the efficiency and accuracy are illustrated by numerical tests.

2016, Gubisch, Martin

In this thesis linear-quadratic optimal control problems for dynamical systems modeled by parabolic partial differential equations with control and state constraints are observed. Different model order reduction techniques basing on a spectral method called proper orthogonal decomposition are analyzed and both a-priori and a-posteriori error bounds are developed to quantify the arising model reduction errors efficiently. Iterative solution techniques for the coupled nonlinear optimality equations are proposed and an associated convergence analysis is provided. The theoretical findings are visualized by numerical tests which illustrate both the advantages and limits of the introduced model reduction strategies.

2014, Gubisch, Martin, Volkwein, Stefan

In this work linear-quadratic optimal control problems for parabolic equations with mixed control-state constraints are considered. These problems arise when a Lavrentiev regularization is utilized for state constrained linear-quadratic optimal control problems. For the numerical solution a Galerkin discretization is applied utilizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined how far the suboptimal control, computed on the basis of the POD method, is from the (unknown) exact one. Numerical examples illustrate the theoretical results. In particular, the POD Galerkin scheme is applied to a problem with state constraints.

2010, Gubisch, Martin

Wir betrachten das System der Magneto-Thermo-Elastizitätsgleichungen mit "second sound" im dreidimensionalen Raum. Zunächst wird das Cauchy-Problem aus den physikalischen Grundgleichungen der Elektrodynamik, der Thermodynamik und den Elastizitätsgesetzen hergeleitet. Mit Hilfe der Theorie der Operatorhalbgruppen wird die Existenz und Eindeutigkeit von Lösungen gezeigt, anschließend werden unter geeigneten Forderungen an die Anfangsdaten polynomiale Abklingraten nachgewiesen. Das Langzeitverhalten des Systems mit "second sound" wird mit dem des klassischen Systems verglichen.

2017, Gubisch, Martin, Volkwein, Stefan

Optimal control problems for partial differential equations (PDEs) are often hard to tackle numerically because their discretization leads to very large scale optimization problems. Therefore, different techniques of model reduction were developed to approximate these problems by smaller ones that are tractable with less effort.

2015, Grimm, Eva, Gubisch, Martin, Volkwein, Stefan

In this work linear-quadratic optimal control problems for parabolic equa- tions with control and state constraints are considered. Utilizing a Lavrentiev regu- larization we obtain a linear-quadratic optimal control problem with mixed control- state constraints. For the numerical solution a Galerkin discretization is applied uti- lizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined by a-posteriori error analysis how far the suboptimal control, com- puted on the basis of the POD method, is from the (unknown) exact one. POD basis updates are computed by optimality-system POD. Numerical examples illustrate the theoretical results for control and state constrained optimal control problems.

2013, Gubisch, Martin, Volkwein, Stefan

In this work linear-quadratic optimal control problems for parabolic equations with mixed control-state constraints are considered. These Problems arise when a Lavrentiev regularization is utilized for state constrained linear-quadratic optimal control problems. For the numerical solution a Galerkin discretization is applied utilizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined how far the suboptimal control, computed on the basis of the POD method, is from the (unknown) exact one. Numerical examples illustrate the theoretical results. In particular, the POD Galerkin scheme is applied to a problem with state constraints.

2017, Gräßle, Carmen, Gubisch, Martin, Metzdorf, Simone, Rogg, Sabrina, Volkwein, Stefan

In the present paper a semilinear boundary control problem is considered. For its numerical solution proper orthogonal decomposition (POD) is applied. POD is based on a Galerkin type discretization with basis elements created from the evolution problem itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. Therefore, different POD basis update strategies which avoids this problem of unmodelled dynamics are com- pared numerically.

2015, Grimm, Eva, Gubisch, Martin, Volkwein, Stefan

In this work linear-quadratic optimal control problems for parabolic equations with control and state constraints are considered. Utilizing a Lavrentiev regularization we obtain a linear-quadratic optimal control problem with mixed control-state constraints. For the numerical solution a Galerkin discretization is applied utilizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined by a-posteriori error analysis how far the suboptimal control, computed on the basis of the POD method, is from the (unknown) exact one. POD basis updates are computed by optimality-system POD. Numerical examples illustrate the theoretical results for control and state constrained optimal control problems.

2011, Gubisch, Martin

We consider the Cauchy problem of magneto-thermo-elasticity with second sound in three space dimensions. After proving the existence of a unique solution, we use Fourier transform and multiplier methods to show polynomial decay rates for suitable initial data. We compare the qualitative and quantitative asymptotic behaviour of magneto-thermo-elasticity with second sound with that of the classical system.