Polynomial Stability of the Timoshenko Beam System with a Fractional Dynamic Boundary Feedback

We investigate the asymptotic behavior of a Timoshenko beam system in which the free end carries a tip mass subject to a restoring spring force and a tempered Caputo fractional damping force with parameter η>0 (the case η=0 is left as an open problem). Using a diffusive state space reformulation of the fractional term, the original problem is embedded into an augmented first-order evolution system on a carefully constructed Hilbert space. Well-posedness is established via the Lumer–Phillips theorem. A spectral analysis of the governing operator, combined with the Arendt–Batty–Lyubich–Vũ theorem, shows that the associated C0-semigroup is strongly asymptotically stable even when the classical equal-wave-speeds condition for the Timoshenko system is violated, provided η>0. Moreover, resorting to the Borichev–Tomilov resolvent method, we reduce the polynomial energy decay to a single resolvent exponent ℓ>0, so that the energy of every solution issued from a datum in the domain of the generator decays at least as fast as t−1/ℓ as t→+∞. We establish the estimates that control ℓ; we identify the mechanism that governs it—the inertia of the tip mass, which screens the damper at high frequency—and we measure ℓ numerically. In particular, the exponent is not dictated by the second-order character of the Timoshenko operator, contrary to what a comparison with the fourth–order beam might suggest.