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Analysis und Systemtheorie


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Prof. Dr. Achim Ilchmann

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Bergmann, Jean Pierre; Bielenin, Martin; Herzog, Roland A.; Hildebrand, Jörg; Riedel, Ilka; Schricker, Klaus; Trunk, Carsten; Worthmann, Karl
Prevention of solidification cracking during pulsed laser beam welding. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (7 Seiten). - (Preprint. - M17,05)

We present a mathematical model to describe laser beam welding based on the heat equation. Since the material coeff cients depend on the temperature, this leads to a quasi-linear parabolic partial differential equation. It is our goal to prevent solidif cation cracking. We address this problem by means of optimal control. It is the intensity prof le of the laser beam which acts as the control function. The main challenge is the formulation of a suitable objective function. In particular, high velocities of the solidif cation interface need to be properly penalized in order to deal with and avoid cracking phenomena.


https://www.db-thueringen.de/receive/dbt_mods_00032771
Gernandt, Hannes; Krauße, Dominik; Sommer, Ralf; Trunk, Carsten
A new method for network redesign via rank one updates. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (5 Seiten). - (Preprint. - M17,05)

We present a method to place the eigenvalues of an electrical network towards a prescribed set of complex numbers by inserting an additional capacitance into the network. We use recent results on rank one perturbations of regular matrix pencils and provide an upper bound on the approximation error of the eigenvalues in the chordal distance.


https://www.db-thueringen.de/receive/dbt_mods_00032770
Worthmann, Karl; Mehrez, Mohamed W.; Mann, George K. I.; Gosine, Raymond G.; Pannek, Jürgen
Interaction of open and closed loop control in MPC. - In: Automatica : a journal of IFAC, the International Federation of Automatic Control. - Amsterdam [u.a.] : Elsevier, Pergamon Press, ISSN 00051098, Bd. 82 (2017), S. 243-250
https://doi.org/10.1016/j.automatica.2017.04.038
Müller, Matthias A.; Worthmann, Karl
Quadratic costs do not always work in MPC. - In: Automatica : a journal of IFAC, the International Federation of Automatic Control. - Amsterdam [u.a.] : Elsevier, Pergamon Press, ISSN 00051098, Bd. 82 (2017), S. 269-277
https://doi.org/10.1016/j.automatica.2017.04.058
Fleige, Andreas; Winkler, Henrik
An indefinite inverse spectral problem of Stieltjes type. - In: Integral equations and operator theory : IEOT. - Berlin : Springer, ISSN 14208989, Bd. 87 (2017), 4, S. 491-514
https://doi.org/10.1007/s00020-017-2358-x
Gernandt, Hannes; Trunk, Carsten
Eigenvalue placement for regular matrix pencils with rank one perturbations. - In: SIAM journal on matrix analysis and applications. - Philadelphia, Pa : Soc, ISSN 10957162, Bd. 38 (2017), 1, S. 134-154
http://dx.doi.org/10.1137/16M1066877
Ilchmann, Achim; Reis, Timo
. - Surveys in differential-algebraic equations ; 4. - Cham : Springer. - ix, 305 Seiten. - (Differential-algebraic equations forum)
http://www.gbv.de/dms/ilmenau/toc/884524973.PDF
Jacob, Birgit; Tretter, Christiane; Trunk, Carsten; Vogt, Hendrik
Numerical range and quadratic numerical range for damped systems. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (27 Seiten). - (Preprint. - M17,05)

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations z'' (t) + D z' (t) + A_0 z(t) = 0 in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as A_0. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients A_0 and D which improve earlier results for sectorial and selfadjoint D; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.


https://www.db-thueringen.de/receive/dbt_mods_00031984
Ilchmann, Achim; Reis, Timo
Outer transfer functions of differential-algebraic systems. - In: Control, optimisation and calculus of variations : COCV. - Les Ulis : EDP Sciences, ISSN 12623377, Bd. 23 (2017), 2, S. 391-425
https://doi.org/10.1051/cocv/2015051
Derkach, Vladimir; Trunk, Carsten
Coupling of definitizable operators in Krein spaces. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (18 Seiten). - (Preprint. - M17,03)

Indefinite Sturm-Liouville operators defined on the real line are often considered as a coupling of two semibounded symmetric operators defined on the positive and the negative half axis. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on the real line.


https://www.db-thueringen.de/receive/dbt_mods_00031469