Phd Degree / Doktora

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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.
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.

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