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This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.
Tian et al. (Sat,) studied this question.