Extending the SCOLE Paradigm to an Interconnected Two-Beam System with Mixed-Order Boundary-Interface Control Strategy

We investigate a SCOLE-type hybrid PDE-ODE system consisting of two Euler-Bernoulli beams serially connected at an interior joint carrying both a point mass and a rotational inertia. The coupling is imposed through dynamic transmission conditions, while each sub-beam preserves classical Euler-Bernoulli dynamics. For a class of boundary-interface velocity feedback laws combining lower-order and higher-order traces at the joint, we establish a complete classification of the system’s asymptotic behavior. When boundary dissipation is active and at least one higher-order interface feedback term is present, the associated C₀ C 0 -semigroup is exponentially stable, without arithmetic restrictions on material parameters, feedback gains, or the interface location. In contrast, if higher-order interface feedback is replaced by lower-order damping alone, exponential stability fails and the total energy decays at the sharp polynomial rate t⁻¹ t - 1 . The proofs rely on sharp frequency-domain resolvent estimates in the Borichev-Tomilov framework, which is shown to be essential due to the failure of classical spectral, multiplier, and Riesz basis methods for this class of interface-inertial generators.