@phdthesis{Sapozhnikova2018,
  author    = {Sapozhnikova, Kateryna},
  title     = {Robust Stability of Differential Equations with Maximum},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-173945},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.},
  subject      = {Differentialgleichung},
  language  = {en}
}
@phdthesis{Schoenlein2012,
  author    = {Sch{\"o}nlein, Michael},
  title     = {Stability and Robustness of Fluid Networks: A Lyapunov Perspective},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-72235},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {In the verification of positive Harris recurrence of multiclass queueing networks the stability analysis for the class of fluid networks is of vital interest. This thesis addresses stability of fluid networks from a Lyapunov point of view. In particular, the focus is on converse Lyapunov theorems. To gain an unified approach the considerations are based on generic properties that fluid networks under widely used disciplines have in common. It is shown that the class of closed generic fluid network models (closed GFNs) is too wide to provide a reasonable Lyapunov theory. To overcome this fact the class of strict generic fluid network models (strict GFNs) is introduced. In this class it is required that closed GFNs satisfy additionally a concatenation and a lower semicontinuity condition. We show that for strict GFNs a converse Lyapunov theorem is true which provides a continuous Lyapunov function. Moreover, it is shown that for strict GFNs satisfying a trajectory estimate a smooth converse Lyapunov theorem holds. To see that widely used queueing disciplines fulfill the additional conditions, fluid networks are considered from a differential inclusions perspective. Within this approach it turns out that fluid networks under general work-conserving, priority and proportional processor-sharing disciplines define strict GFNs. Furthermore, we provide an alternative proof for the fact that the Markov process underlying a multiclass queueing network is positive Harris recurrent if the associate fluid network defining a strict GFN is stable. The proof explicitely uses the Lyapunov function admitted by the stable strict GFN. Also, the differential inclusions approach shows that first-in-first-out disciplines play a special role.},
  subject      = {Warteschlangennetz},
  language  = {en}
}
@phdthesis{Kessler2000,
  author    = {Keßler, Manuel},
  title     = {Die Ladyzhenskaya-Konstante in der numerischen Behandlung von Str{\"o}mungsproblemen},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-2791},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2000},
  abstract  = {Charakteristisch f{\"u}r die L{\"o}sbarkeit von elliptischen partiellen Differentialgleichungssystemen mit Nebenbedingungen ist das Auftreten einer inf-sup-Bedingung. Im prototypischen Fall der Stokes-Gleichungen ist diese auch als Ladyzhenskaya-Bedingung bekannt. Die G{\"u}ltigkeit dieser Bedingung, bzw. die Existenz der zugeh{\"o}rigen Konstante ist eine Eigenschaft des Gebietes, innerhalb dessen die Differentialgleichung gel{\"o}st werden soll. W{\"a}hrend die Existenz schon die L{\"o}sbarkeit garantiert, ist beispielsweise f{\"u}r Fehleraussagen bei der numerischen Approximation auch die Gr{\"o}ße der Konstanten sehr wichtig. Insbesondere auch deshalb, weil eine {\"a}hnliche inf-sup-Bedingung auch bei der Diskretisierung mittel Finiter-Elemente-Methoden auftaucht, die hier Babuska-Brezzi-Bedingung heißt. Die Arbeit befaßt sich auf der einen Seite mit einer analytischen Absch{\"a}tzung der Ladyzhenskaya-Konstante f{\"u}r verschiedene Gebiete, wobei {\"A}quivalenzen mit verwandten Problemen aus der komplexen Analysis (Friedrichs-Ungleichung) und der Strukturmechanik (Kornsche Ungleichung) benutzt werden. Ein weiterer Teil befaßt sich mit dem Zusammenhang zwischen kontinuierlicher Ladyzhenskaya- Konstante und diskreter Babuska-Brezzi-Konstante. Die dabei gefundenen Ergebnisse werden mit Hilfe eines dazu entwickelten leistungsf{\"a}higen Finite-Elemente-Programmsystems numerisch verifiziert. Damit k{\"o}nnen erstmals genaue Absch{\"a}tzungen der Konstanten in zwei und drei Dimensionen gefunden werden. Aufbauend auf diesen Resultaten wird ein schneller L{\"o}sungsalgorithmus f{\"u}r die Stokes-Gleichungen vorgeschlagen und anhand von problematischen Gebieten dessen {\"U}berlegenheit gegen{\"u}ber klassischen Verfahren wie beispielsweise der Uzawa-Iteration demonstriert. W{\"a}hrend selbst bei einfachen Geometrien eine Konvergenzbeschleunigung um einen Faktor 5 erwartet werden kann, sind in kritischen F{\"a}llen Faktoren bis zu 1000 m{\"o}glich.},
  subject      = {Stokes-Gleichung},
  language  = {de}
}
@misc{Proell2013,
  type      = {Master Thesis},
  author    = {Pr{\"o}ll, Sebastian},
  title     = {Stability of Switched Epidemiological Models},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-108573},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2013},
  abstract  = {In this thesis it is shown how the spread of infectious diseases can be described via mathematical models that show the dynamic behavior of epidemics. Ordinary differential equations are used for the modeling process. SIR and SIRS models are distinguished, depending on whether a disease confers immunity to individuals after recovery or not. There are characteristic parameters for each disease like the infection rate or the recovery rate. These parameters indicate how aggressive a disease acts and how long it takes for an individual to recover, respectively. In general the parameters are time-varying and depend on population groups. For this reason, models with multiple subgroups are introduced, and switched systems are used to carry out time-variant parameters. When investigating such models, the so called disease-free equilibrium is of interest, where no infectives appear within the population. The question is whether there are conditions, under which this equilibrium is stable. Necessary mathematical tools for the stability analysis are presented. The theory of ordinary differential equations, including Lyapunov stability theory, is fundamental. Moreover, convex and nonsmooth analysis, positive systems and differential inclusions are introduced. With these tools, sufficient conditions are given for the disease-free equilibrium of SIS, SIR and SIRS systems to be asymptotically stable.},
  subject      = {Gew{\"o}hnliche Differentialgleichung},
  language  = {en}
}
@phdthesis{Suttner2020,
  author    = {Suttner, Raik},
  title     = {Output Optimization by Lie Bracket Approximations},
  doi       = {10.25972/OPUS-21177},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-211776},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {In this dissertation, we develop and analyze novel optimizing feedback laws for control-affine systems with real-valued state-dependent output (or objective) functions. Given a control-affine system, our goal is to derive an output-feedback law that asymptotically stabilizes the closed-loop system around states at which the output function attains a minimum value. The control strategy has to be designed in such a way that an implementation only requires real-time measurements of the output value. Additional information, like the current system state or the gradient vector of the output function, is not assumed to be known. A method that meets all these criteria is called an extremum seeking control law. We follow a recently established approach to extremum seeking control, which is based on approximations of Lie brackets. For this purpose, the measured output is modulated by suitable highly oscillatory signals and is then fed back into the system. Averaging techniques for control-affine systems with highly oscillatory inputs reveal that the closed-loop system is driven, at least approximately, into the directions of certain Lie brackets. A suitable design of the control law ensures that these Lie brackets point into descent directions of the output function. Under suitable assumptions, this method leads to the effect that minima of the output function are practically uniformly asymptotically stable for the closed-loop system. The present document extends and improves this approach in various ways. One of the novelties is a control strategy that does not only lead to practical asymptotic stability, but in fact to asymptotic and even exponential stability. In this context, we focus on the application of distance-based formation control in autonomous multi-agent system in which only distance measurements are available. This means that the target formations as well as the sensed variables are determined by distances. We propose a fully distributed control law, which only involves distance measurements for each individual agent to stabilize a desired formation shape, while a storage of measured data is not required. The approach is applicable to point agents in the Euclidean space of arbitrary (but finite) dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic (and even exponential) stability. A similar statement is also derived for nonholonomic unicycle agents with all-to-all communication. We also show how the findings can be used to solve extremum seeking control problems. Another contribution is an extremum seeking control law with an adaptive dither signal. We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. A key novelty of the approach is an adaptive choice of the frequency parameter. In this way, the task of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of a minimum, but leads to exact convergence. For the case of an output function with a global minimum and no other critical point, we prove global convergence. Finally, we present an extremum seeking control law for a class of nonholonomic systems. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. This requires a two-fold approximation of Lie brackets on different time scales. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent space. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.},
  subject      = {Extremwertregelung},
  language  = {en}
}