Numerous systems with bang-bang control algorithms and mechanical systems with dry friction operate in oscillatory modes. Such systems include, for example, fuel injection control systems [1], in which the sensor shows only the sign of the deviation of the exhaust parameters from given values and the basic operation mode is oscillations in a neighborhood of the given values. Another example of such systems is given by oscillations of a pendulum on an inclined uniformly rotating disk in the presence of dry friction [2]. Such motions have the specific feature that, for some part of the period, the pendulum moves together with the disk (in a sliding mode) and then returns to the original position. At the same time, real control systems always contain actuators, sensors, and other devices whose operation is described by differential equations with small parameters multiplying the derivatives (these parameters correspond to the time constants of these devices); consequently, the complete model of such a system is described by singularly perturbed bang-bang systems (SPBBS). In fuel injection control systems [1], SPBBS can describe, say, the influence of the engine on the injector operation. In the system with a rotating disk, dealing with SPBBS is necessitated if one considers the problem of perturbations induced by an additional pendulum elastically connected to the pendulum lying on the disk in the case of dry friction [2]. It was shown in [3–5] that slow periodic solutions of smooth singularly perturbed systems lie on slow integral manifolds. The integral manifold of slow motions of a SPBBS consists of leaves corresponding to different values of the bang-bang control. We show that slow periodic solutions of SPBBS have interior boundary layers appearing in the transition from a neighborhood of one leaf of the integral manifold into a neighborhood of another. Slow periodic solutions of SPBBS are also characterized by the fact that, unlike relaxation oscillations (see the bibliography in [6]), the right-hand sides of the equations describing rapid motions are nonzero at control switching points, which permits one to use the boundary function method [7] for describing the breakaway of a solution from a neighborhood of one leaf of the integral manifold into a neighborhood of another leaf.
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