Tez Türü: Doktora
Tezin Yürütüldüğü Kurum: Yıldız Teknik Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Bölümü, Türkiye
Tez Danışmanı: Eyüp Kızıl
Tezin Onay Tarihi: 2024
Tezin Dili: İngilizce
Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
Özet:
The thesis focuses primarily on the study of linear control systems on homogeneous spaces, in particular in the context of the 3D Heisenberg Lie group $\HB$ and its closed subgroups. The main motivation stems from a result of P. Jouan \cite{Jouan}, which shows an interesting connection between control affine systems on manifolds and linear control systems (abbreviated as LCSs) on Lie groups or homogeneous spaces. The goal is to provide a comprehensive characterization of all possible LCSs on homogeneous spaces of $\HB$ and to investigate their dynamical properties. The comprehensive analysis covers various dimensions of closed subgroups, providing a thorough understanding of LCSs on homogeneous spaces of $\HB$. The first chapters lay the foundation by introducing basic concepts related to linear vector fields, LCSs on Lie groups, and the classification of closed subgroups of $\HB$.
Building on this, subsequent chapters deal with the projection of LCSs onto homogeneous spaces, considering invariance criteria for subgroups under the flows of systems. The core analysis revolves around controllability and control sets, with detailed investigations of non-normal subgroups of different dimensions. A special attention is given to the more complex structures and controllability issues of certain cases. Each case is studied in detail, step by step, and the geometric descriptions obtained are given, with emphasis on their topological and geometric properties. In the initial chapters, we provide an overview of fundamental concepts in theory, aiming to enhance comprehension of the subsequent sections of the thesis. Chapters \ref{LCStan} and \ref{lcsofhom} focus on constructing the necessary results for explicitly characterizing all possible LCSs on the homogeneous spaces of the Heisenberg group. Chapter \ref{lastsec} deals with the dynamics of these systems and provides a detailed analysis of their controllability issues. In the final chapter, the results obtained so far are reviewed and the implications of the findings are also presented, along with the consideration of new problems to be addressed in future research.