A coupled elastic beam system with a controller at the hinged junction that has point-mass is formulated as an abstract evolution equation in the energy space. By the spectral analysis of the hybrid differential operators with mixed boundary-junction conditions, an alternative principle of stabilizability in terms of the beam lengths ρ1 and ρ2 is proved: (I) If ρ1/ρ2 equals a quotient of two positive roots (different or same) of the transcendental equation tanμ = tanhμ, then the pointwise stabilization of the system is impossible. (II) If ρ1/ρ2 does not equal any quotient described above, then the system is strongly stabilized in the energy space by a pointwise damping feedbackf(t) = −wt(ρ1,t) at the junction point x = ρ1. Furthermore in the first case it is proved that a combination of the above pointwise damping feedback and an appropriate quasi-pointwise damping feedback on one beam achieves the strong stabilization.