Topological phases such as polar skyrmions have been a fertile playground for ferroelectric oxide superlattices, with exotic physical phenomena such as negative capacitance. Herein, using phase-field simulations, we demonstrate the local control of the skyrmion phase with electric potential applied through a top electrode. Under a relatively small electric potential, the skyrmions underneath the electrode can be erased and recovered reversibly. A topologically protected transition from the symmetric to asymmetric skyrmion bubbles is observed at the edge of the electrode. While a topological transition to a labyrinthine domain requires a high applied potential, it can switch back to the skyrmion state with a relatively small electric potential. The topological transition from +1 to 0 occurs before the full destruction of the bubble state. It is shown that the shrinking and bursting of the skyrmions leads to a large reduction in the dielectric permittivity, the magnitude of which depends on the size of the electrode.
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