Abstract:
Impulsive systems have become increasingly popular in recent decades because they provide a natural toolkit for mathematical modeling of many phenomena in the real world and industrial processes. Applications of impulsive systems can be found in a wide variety of fields, such as aeronautics, environmental science, economics, epidemiology, finance, medicine, and robotics, to name a few. Solutions of impulsive systems are discontinuous and this makes some standard methods of analysis and control ineffective. However, much progress has been made in recent years, and many interesting results in stability, manifold theory and bifurcation analysis have been published for such systems. Extensive literature is devoted to the stability conditions for solutions of impulsive systems. But the critical case of stability has not yet been sufficiently studied. The paper concerns in periodic impulsive systems with fixed moments of impulse actions. Such a system consists of two elements: a continuous system of differential equations that governs the state of the system between moments of impulsive events, and a system of difference equations, which specifies a method for instantly changing the state of the system. The critical case of stability is considered for a periodic impulsive system of the second order of general form with an autonomous differential equation and an autonomous impulsive action operator. It is assumed that the linear approximation monodromy matrix has a pair of complex conjugate multipliers on the unit circle of the complex plane. The algorithm for calculating the first Lyapunov quantity is described in detail. The reliability of the algorithm formulas is confirmed by the results of test analytical calculations for two illustrative examples.
Keywords:periodic impulsive systems, reduction to diskrete time, critical case of stability, first Lyapunov quantity, monodromy matrix, complex conjugate multipliers.
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