This paper considers the equations of motion of mechanical systems subject to inequality constraints, which can be obtained by looking for the stationary value of the action integral, Two different methods are used to take into account the inequality constraints in the computation of the stationary value of the action integral: the method of the Valentine variables and the method of the penalty functions. The equations of motion resulting from the application of the method of the Valentine variables, which introduces the concept of "nonsmooth" impacts, constitute the exact model of the constrained mechanical system; such a model is suitable to be employed when the impacting parts of the actual mechanical system are very stiff. The equations of motion resulting from the application of the method of the penalty Functions, which introduces the concept of "smooth impacts," constitute an approximate model of the constrained mechanical system; such a model is suitable to be employed when the impacting parts of the actual mechanical system show some flexibility. Various feedback control laws from the natural outputs and from their time derivatives are studied with reference to both models of impact; the closed-loop systems resulting from the application of the same control law to both models show pretty much the same global asymptotic stability properties. The proposed control laws are only concerned with regulation problems in the presence of possible contacts and impacts among parts of the mechanical system or with the external environment; the problem of controlling these mechanical systems along time-varying trajectories is not considered in this paper. The effectiveness of the proposed control structure has been tested experimentally with reference to a single-link robot arm, showing a valuable behavior.