Speaker. Kais Ammari, University of Monastir, Tunisia.

Title. Stability of some delayed systems.

Abstract. In this presentation, we address a stabilization problem for a generalized thermoelastic system with delay, commonly referred to as the $\alpha-\beta$ system. Thermoelastic systems describe the interaction between mechanical deformations and thermal effects in elastic materials, and the inclusion of delay terms introduces additional challenges in the analysis of their stability. Delays in such systems can arise from physical phenomena such as heat conduction or time lags in control mechanisms, making them particularly relevant in practical applications. To analyze this system, we first establish its well-posedness by utilizing the semigroup approach. Specifically, we show that the $\alpha-\beta$ system generates a C$_0$-semigroup, ensuring the existence, uniqueness, and continuous dependence of the solutions on the initial data.  Next, we turn to the stability analysis, focusing on both exponential and polynomial stability under certain conditions. By employing a frequency-domain approach, we derive conditions that ensure the system's response diminishes over time, either at an exponential rate or at a polynomial rate.  Finally, we illustrate these theoretical results by applying them to specific examples within the field of thermoelasticity. These examples highlight the practical implications of the stabilization techniques and demonstrate how the results can be applied to real systems.