Optimal Polynomial Stability of the Porous Elastic System with One Dynamic Boundary Control

This paper investigates the asymptotic stability of a one-dimensional porous elastic system subject to a single boundary control of the fractional derivative type. The system consists of two coupled hyperbolic equations describing the displacement of an elastic solid and the volume fraction, with boundary conditions that include a generalized Caputo fractional derivative of order α∈(0,1) at x=L. This configuration has not been previously addressed in the literature. Using semigroup theory, we first reformulate the system as an abstract Cauchy problem and prove that the associated operator generates a C0-semigroup of contractions on a suitable energy space, establishing global well-posedness. Under explicit and generic conditions on the physical parameters and the length L, we prove strong stability via the Arendt–Batty criterion, showing that all solutions tend to zero in the energy norm as t→∞. The main result is a polynomial decay rate: there exists c>0 such that ∥SA(t)U0∥H≤ct−1/6∥U0∥D(A) for all initial data in the domain of the generator. The proof relies on the Borichev–Tomilov theorem and a detailed contradiction argument based on asymptotic expansions of the resolvent operator. Notably, the decay rate is independent of any relation between the wave propagation speeds, which distinguishes our result from many previous studies on porous elastic or Timoshenko systems.