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This paper addresses control of structural vibrations using semi-active actuators that are capable of producing variable stiffness. Usually vibration suppression is achieved using damping devices rather than variable stiffness ones. However, variable stillness devices have significant advantages for shock isolation purposes. In this work we use a passivity approach to establish the requirements for the control law for a structure equipped with semi-active variable stiffness devices. We also solve a minimum-time, optimal control problem that demonstrates that our, passive, resetting feedback control law approximates the optimal control. An example simulation is given that shows favorable damping qualities are achieved for a three story building subject to the El Centro earthquake.