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  <doc>
    <id>214</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2017-11-30</completedDate>
    <publishedDate>2017-11-20</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Numerical optimal control of instationary gas transport with control and state constraints</title>
    <abstract language="eng">We consider the optimal control of a nonlinear hyperbolic system of balance&#13;
laws on a one-dimensional network which arises in the context of gas transport in pipeline&#13;
systems. State constraints, which are required for the safe operation of the system, are&#13;
incorporated by a barrier method. We discuss the well-posedness of the governing system&#13;
of partial differential-algebraic equations and investigate the existence of minimizers. For&#13;
the numerical solution, we then consider the approximation of the state equation by&#13;
mixed finite elements in space and a particular linear implicit time integration scheme&#13;
that can be interpreted as a discontinuous Galerkin approximation. We establish well-&#13;
posedness of this discretization scheme and prove the existence of minimizers for the&#13;
corresponding discretized optimal control problem and discuss its numerical solution&#13;
by a projected Gauß-Newton method. The efficient realization of the Jacobian and&#13;
Hessian of the quadratic approximations that have to be minimized in every iteration&#13;
of the Gauß-Newton method can be obtained via the solution of discretized sensitivity&#13;
and adjoint equations. These are obtained by formal differentiation and transposition&#13;
of the Galerkin methods employed for the discretization of the state equations. All&#13;
approximations obtained after discretization can thus be interpreted as functions on the&#13;
continuous level and, since the functional analytic setting is not changed by the Galerkin&#13;
discretization, we observe mesh independence of the resulting fully discrete methods. For&#13;
illustration of our theoretical results and to demonstrate the efficiency of the proposed&#13;
method, we present numerical results for two test problems that model typical situations&#13;
that may arise in the daily operation of gas networks.</abstract>
    <enrichment key="review.accepted_by">2</enrichment>
    <author>Herbert Egger</author>
    <author>Thomas Kugler</author>
    <author>Winnifried Wollner</author>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="subprojects" number="">C04</collection>
    <collection role="subprojects" number="">A08</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/214/optimalcontrol.pdf</file>
  </doc>
  <doc>
    <id>207</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2017-01-15</completedDate>
    <publishedDate>2017-11-20</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">An asymptotic preserving mixed finite element method for wave propagation in pipelines</title>
    <abstract language="eng">We consider a parameter dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation we then consider a mixed finite element method in space together with a Runge-Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general.</abstract>
    <enrichment key="review.accepted_by">2</enrichment>
    <author>Herbert Egger</author>
    <author>Thomas Kugler</author>
    <collection role="subprojects" number="">C04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/207/asymptotic.pdf</file>
  </doc>
  <doc>
    <id>148</id>
    <completedYear>2017</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2017-11-04</completedDate>
    <publishedDate>2017-07-20</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On structure preserving model reduction for damped wave propagation in transport networks</title>
    <abstract language="eng">We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-&#13;
discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic&#13;
equations while the intermediate discretization by finite elements is only required for the actual&#13;
computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <author>Herbert Egger</author>
    <author>Thomas Kugler</author>
    <author>Björn Liljegren-Sailer</author>
    <author>Nicole Marheineke</author>
    <author>Volker Mehrmann</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>partial differential-algebraic equations</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>port-Hamiltonian systems</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Galerkin projection</value>
    </subject>
    <collection role="institutes" number="">Friedrich-Alexander-Universität Erlangen-Nürnberg</collection>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="institutes" number="">Technische Universität Berlin</collection>
    <collection role="subprojects" number="">B03</collection>
    <collection role="subprojects" number="">C02</collection>
    <collection role="subprojects" number="">C04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/148/EggKLMM17.pdf</file>
  </doc>
  <doc>
    <id>78</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2016-09-16</completedDate>
    <publishedDate>2016-09-16</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Uniform exponential stability of Galerkin approximations for damped wave systems</title>
    <abstract language="eng">We consider the numerical approximation of linear damped wave systems&#13;
by Galerkin approximations in space and appropriate time-stepping schemes. Based on&#13;
a dissipation estimate for a modified energy, we prove exponential decay of the physical&#13;
energy on the continuous level provided that the damping is effective everywhere in the&#13;
domain. The methods of proof allow us to analyze also a class of Galerkin approximations&#13;
based on a mixed variational formulation of the problem. Uniform exponential stabil-&#13;
ity can be guaranteed for these approximations under a general compatibility condition&#13;
on the discretization spaces. As a particular example, we discuss the discretization by&#13;
mixed finite element methods for which we obtain convergence and uniform error esti-&#13;
mates under minimal regularity assumptions. We also prove unconditional and uniform&#13;
exponential stability for the time discretization by certain one-step methods. The valid-&#13;
ity of the theoretical results as well as the necessity of some of the conditions required&#13;
for our analysis are demonstrated in numerical tests</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <author>Herbert Egger</author>
    <author>Thomas Kugler</author>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="subprojects" number="">C04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/78/damped.pdf</file>
  </doc>
  <doc>
    <id>79</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2016-09-16</completedDate>
    <publishedDate>2016-09-16</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Damped wave systems on networks: exponential stability and uniform approximations</title>
    <abstract language="eng">We consider a damped linear hyperbolic system modelling the propagation&#13;
of pressure waves in a network of pipes. Well-posedness is established via semi-group&#13;
theory and the existence of a unique steady state is proven in the absence of driving&#13;
forces. Under mild assumptions on the network topology and the model parameters,&#13;
we show exponential stability and convergence to equilibrium. This generalizes related&#13;
results for single pipes and multi-dimensional domains to the network context. Our proof&#13;
of the exponential stability estimate is based on a variational formulation of the problem,&#13;
some graph theoretic results, and appropriate energy estimates. The main arguments&#13;
are rather generic and can be applied also for the analysis of Galerkin approximations.&#13;
Uniform exponential stability can be guaranteed for the resulting semi-discretizations&#13;
under mild compatibility conditions on the approximation spaces. A particular realiza-&#13;
tion by mixed finite elements is discussed and the theoretical results are illustrated by&#13;
numerical tests in which also bounds for the decay rate are investigated.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <author>Herbert Egger</author>
    <author>Thomas Kugler</author>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="subprojects" number="">C04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/79/network.pdf</file>
  </doc>
</export-example>
