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Electrical tuning the radiative damping of "nonradiative" (weak) terahertz plasmon mode in a periodic dual-grating-gate graphene structure is studied theoretically. Tuning the dc voltage value applied to one of the gate subgratings can change the radiative damping of the weak plasmon mode from zero to the dissipative damping of the plasmons in high-quality graphene allowing for maximum energy conversion from the incident wave to "nonradiative" plasmon mode. Radiative damping increases due to the increase of the dipole moment of this mode by virtue of its asymmetrical gating. Using the weak plasmon modes is important for effective excitation of the plasmon resonances in high-quality graphene as well as for realizing the concept of low-threshold terahertz lasers based on graphene plasmonic structures. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. [ABSTRACT FROM AUTHOR]