Energy Decay of Solutions to the Euler–Bernoulli Beam Equation With A Fractional Derivative Type Dynamic Boundary Control Condition

ABSTRACT We investigate the Euler–Bernoulli beam equation subject to a dynamic boundary control condition involving a Caputo‐type fractional derivative. By using semigroup theory and the Borichev–Tomilov resolvent condition, we establish the well‐posedness of the system and analyze its stability properties. In particular, we prove that the associated semigroup is strongly stable but not uniformly exponentially stable, and we derive a polynomial energy decay rate that depends on the fractional order.