On the exponential stabilization of a semilinear Timoshenko beam system via locally distributed delayed damping

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Abstract
In this paper, we study a semilinear Timoshenko beam system posed in one dimensional inhomogeneous medium. This study focuses on the stability analysis of such a problem under the effect of two localized delayed damping mechanisms. Despite the presence of a time delay, we are able to prove that the system is well-posed in the sense of semigroups theory. Unlike previous works, we show that the exponential decay of the energy of our system can be ensured by a locally distributed delayed damping alone. This result is established under the usual conditions on the damping parameters, namely that the weights of the time delays are smaller than those of the corresponding dampings, regardless of the wave propagation speeds. The proof of our main stability result relies on some techniques from the microlocal analysis theory.

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