Periodic arrays of shunted, piezoelectric patches are employed to control waves propagating over the surface of plate structures, and corresponding vibrations. The shunted, piezoelectric patches act as sources of impedance mismatch, which gives rise to interference phenomena resulting from the interaction between incident, reflected and transmitted waves. Periodically distributed mismatch zones, i.e., the piezo patches, produce frequency dependent, wave-dynamic characteristics, which include the generation of band gaps, or stop bands in the frequency domain. The extent of induced band gaps depends on the mismatch in impedance generated by each patch. The total impedance mismatch, in turn, is determined by the added mass and stiffness of each patch as well as the shunting electrical impedance. Proper selection of the shunting electric-circuit thus provides control over the attenuation capabilities of the piezo-plate structure, as well as the ability to adapt to changing excitation conditions. Control of wave-propagation attenuation and vibration reduction for plates with periodic, shunted, piezoelectric patches is demonstrated numerically, employing finite-element models of the considered structures.