Effect of Geometric Nonlinearity on Stability Results of Extensible Two Beams With Time Fractional Delays

In this article, we analyze the effect of geometric nonlinearity on stability results for extensible nonsymmetric two beams in a bounded domain. Two internal frictional dampings are present together with two fractional time delays. The equations describe a system consisting of two elastically coupled extensible beams, to which uniform damping terms and axial compression loads are applied. We consider an augmented model by means of change of variables. The existence and uniqueness of the solutions are demonstrated by the use of classical semigroup theory, subject to a smallness condition on the fractional delay. Then, using the Lyapunov method, we prove the exponential stability of the augmented model for the nonlinear problem (i.e., with geometric nonlinearity). Furthermore, we prove that the linear problem (i.e., without geometric nonlinearity) is also exponentially stable by employing a frequency domain approach.