This paper deals with the solution and asymptotic analysis for a porous-elastic system with two dynamic control boundary conditions of fractional derivative type. We consider an augmented model. The energy function is presented, and the dissipative property of the system is established. We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem. We present two results for the asymptotic behavior: Strong stability of the $C_0$-semigroup associated with the system using Arendt-Batty and Lyubich-V˜u's general criterion and polynomial stability applying Borichev-Tomilov's Theorem.