Phd Degree / Doktora

Permanent URI for this collectionhttps://hdl.handle.net/11147/2869

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Author: search.filters.author.Batal, Ahmet

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  • Doctoral Thesis
    Initial-Boundary Value Problem for the Higher-Order Nonlinear Schrödinger Equation on the Half-Line
    (01. Izmir Institute of Technology, 2024) Alkın, Aykut; Batal, Ahmet; Özsarı, Türker
    We establish local well-posedness in the sense of Hadamard for the higher-order nonlinear Schrödinger equation with a general power nonlinearity formulated on the halfline {x > 0}. We consider separately the two different scenarios of associated coefficients such that only one boundary condition is required, or exactly two boundary conditions are required. We assume a general nonhomogeneous boundary datum of Dirichlet type at x = 0 for the former case, and we add the Neumann type for the latter case. Our functional framework centers around fractional Sobolev spaces Hs x(R+) with respect to the spatial variable. We treat both high regularity (s > 1 2 ) and low regularity (s < 1 2 ) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initialboundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data.
  • Doctoral Thesis
    Analysis of Damped and Viscoelastic Linear Wave Equations Exposed To External Neumann Manipulations
    (01. Izmir Institute of Technology, 2024) Susuzlu, İdem; Batal, Ahmet; Özsarı, Türker
    In this thesis, the main aim is to study the global existence and the stabilization of solutions for linear damped and viscoelastic wave equations evolving on a bounded medium in an arbitrarily large time interval subject to inhomogeneous Neumann manipulation on a part of the boundary. The analysis of these models reveals additional interesting features and challenges in comparison to their homogeneous counterparts, on which there are studies in the literature. This is due to the fact that, in the present context, the rate at which energy of solution is changed has a dependency on the boundary trace of temporal derivative. It is not clear how this quantity could be controlled in terms of given data according to Sobolev trace theory. Nevertheless, we achieve to establish global existence of solutions first using dynamic extension method to homogenize boundary conditions. Next, we construct the weak solutions of the homogenized models. For the damped wave equation, we rely on the semigroup approach while for the viscoelastic model we use Faedo-Galerkin method. The global unique solutions of the original models are obtained through a reunification argument. Then, we also prove uniform stabilization of solutions with decay rates characterized by the decay behavior of Neumann input using the multiplier (energy) technique. The latter requires a subtle analysis of boundary integrals in energy estimates involving unknown trace terms. We also develop numerical solutions of the models. For the damped wave equation, we rely on the explicit method while for the viscoelastic model we use the Crank-Nicolson method. We support our theoretical result with numerical simulations satisfying given assumptions. We supplement these with further numerical simulations in which data do not necessarily satisfy the given assumptions for decay. The latter offers, at the numerical level, essential physical insights into how energy might change in the presence of, for instance, improper boundary data.
  • Doctoral Thesis
    Stability Analysis of Nonlinear Dynamical Systems With Lévy Typeperturbations
    (01. Izmir Institute of Technology, 2023) Tamcı, Ege; Batal, Ahmet; Savacı, Ferit Acar
    In order to model the noise in power networks, generally, normal distribution is used. However, normal distribution is not convenient in modelling noise which has sudden peaks. Instead, combination of a continuous process and a jump processes is much more suitable. With this idea in mind, in this thesis, the stability and control of two equations used in modeling power grids is analyzed, under the assumption that they are exposed to Lévy process noise which includes jumps. These equations are the swing equation and the Kuramoto Model. The swing equation is used to model the single machine infinite bus system (SMIBS). Kuramoto Model is used when a large number of generators are considered as a network of coupled oscillators with their own natural frequencies. In our stability control study in the SMIBS, the noise in the system has sudden and finite changes is assumed and therefore should be modelled with a modified tempered α-stable process obtained by adding a finite jump condition on the tempered α-stable process when α < 1. The control functions depending on the mechanical power input and damping parameters are designed in order to make the system stable in probability and exponential stable at its equilibrium point. These theoretical results are supported by numerical studies. For Kuromato model, assuming that the power network consists of two layers, namely oscillator, and control layers and that is affected with a general Lévy process which has finite jumps, functions which provide the stability of phase and frequencies are obtained, depending on oscillator and coupling strengths. In the light of the numerical studies, the control of frequency and phase synchronization up to a certain noise intensity level can be evaluated, but it is not possible beyond that level is concluded.
  • Doctoral Thesis
    Boundary Feedback Stabilization of Some Evolutionary Partial Differential Equations
    (01. Izmir Institute of Technology, 2022) Yılmaz, Kemal Cem; Batal, Ahmet; Özsarı, Türker; Özsarı, Türker; Batal, Ahmet
    The purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.