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Hysteresis poses a challenge for control of smart actuators. A fundamental approach to hysteresis control is inverse compensation. For practical implementation, it is desirable for the input function generated via inversion to have regularity properties stronger than continuity. In this paper, we consider the problem of constructing right inverses for the Preisach model for hysteresis. Under mild conditions on the density function, we show the existence and weak-star continuity of the right-inverse, when the Preisach operator is considered to act on Holder continuous functions. Next, we introduce the concept of regularization to study the properties of approximate inverse schemes for the Preisach operator. Then, we present the fixed point and closest-match algorithms for approximately inverting the Preisach operator. The convergence and continuity properties of these two numerical schemes are studied. Finally, we present the results of an open-loop trajectory tracking experiment for a magnetostrictive actuator.