Approximation of Pulse Sliding Modes of Differential Inclu­sions

Differential inclusions with pulse influences are investigated. The basic attention is given to dynamic objects with pulse positional control, which is understood as some abstract operator with Dirac's function (“a running pulse”), concentrated in each moment of time. “Running pulse” as the generalized function has no sense. Its formalization consists in digitization of correcting pulse influences on the system, corresponding to directed set of partitions for an interval of control . Reaction of system to such control are discontinuous movements, which are a network “Euler's broken lines”. In problems of control the special place is occupied the situation when as a result of the next correction the phase point of object appears on some surface. Then at reduction of time between corrections in system the effect such as “slidings” is brought and the network “Euler's broken lines” refers to as a pulse sliding mode. In practical use of procedure of pulse control inevitably there is a problem on replacement of Dirac's pulse to sequence of its continuous approximations of delta-like function. In given article for differential inclusions with positional pulse control in the right-hand part are considered two types of limiting transition on delta-like functions resulting to “Euler's broken lines” and to pulse sliding modes One of them leads to known conditions of an admissibility jump at the moment of pulse influences, and another — determines size of pulse correction directly on value of preset intensity of a pulse depending on time and a condition of object. Researches base on continuous Yosida's approximations of multiple-valued maps and the known facts for the differential equations with pulses.