Strong Stability of a Delayed Viscoelastic Timoshenko System With Fractional Time Delay and Two Fractional Boundary Controls

The strong stability of the Timoshenko beam in viscoelastic systems with a fractional time delay and fractional boundary controls is a complex yet fascinating area of study. This work aims to investigate a mathematical model wherein nontrivial methods are used to solve for both quantitative and qualitative properties. This paper considers a particular Timoshenko system in viscoelastic systems with a fractional time delay and fractional boundary controls. Under assumptions about the data, we establish the well-posedness of the solution and the strong stability. The paper addresses the question of how the constraints used in the boundary in the fractional type affect the stability of our model in the presence of a fractional delay. Proving strong stability in such systems requires a careful analysis of energy decay, taking into account the interaction between fractional delays and boundary feedback. The semigroup approach will be used to show the well-posedness, while energy methods, along with the fractional Sobolev embedding, play a critical role in the proof.