Boundary controllability for a non-autonomous degenerate Timoshenko beam model

This paper investigates, for the first time, the boundary controllability of a non-autonomous degenerate Timoshenko beam. To fully capture the physical reality of modern structures, it is essential to model beam dynamics in environments characterized by both spatial inhomogeneities and temporal variations. Unlike autonomous models, the proposed framework incorporates time-dependent wave propagation speeds, providing a more accurate spatio-temporal representation of realistic wave behavior in structures where parameters evolve under fluctuating environmental or material conditions. We first prove the well-posedness of the associated adjoint problem in appropriate weighted energy spaces, using semigroup theory and Kato's theorem. An observability estimate is then established for the adjoint system, assuming a sufficiently large time T that accounts for the non-autonomous nature and the degeneracy of the model. Combining this estimate with the transposition method, we conclude that the original system is boundary null controllable using the Hilbert Uniqueness Method (HUM). Furthermore, we provide an explicit lower bound for the controllability time, which highlights the intricate interplay between the degree of degeneracy and the coupling dynamics of the shear and bending waves under time-dependent propagation.