Approximation of positional impulse controls for differential inclusions

Nonlinear control systems presented as differential inclusions with positional impulse controls are investigated. By such a control we mean some abstract operator with the Dirac function concentrated at each time. Such a control (“running impulse”), as a generalized function, has no meaning and is formalized as a sequence of correcting impulse actions on the system corresponding to a directed set of partitions of the control interval. The system responds to such control by discontinuous trajectories, which form a network of so-called “Euler's broken lines.” If, as a result of each such correction, the phase point of the object under study is on some given manifold (hypersurface), then a slip-type effect is introduced into the motion of the system, and then the network of “Euler's broken lines” is called an impulse-sliding mode. The paper deals with the problem of approximating impulse-sliding modes using sequences of continuous delta-like functions. The research is based on Yosida's approximation of set-valued mappings and some well-known facts for ordinary differential equations with impulses.