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A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected π kinks and of an individual π kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.Ministerio de Economía y Competitividad (Spain), grants FQM262  FQM207 (N.R.Q.), FQM-7276P09-FQM-4643 (N.R.Q., R.A.N.) from the Junta de Andalucía (Spain

Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models