Introduction

Abstract

The concept of a discrete-continuous or hybrid system (DCS or HS) arose in scientific practice on the boundary of 50-60-th years. Its appearance was connected with the development of digital automatic controls for continuous plants [199], [200]. These systems usually have the continuous and discrete (impulsive) parts which operate together providing the new properties to the system. Since then it had become clear that the presence of elements operating in a discrete or impulsive mode may lead to substantial changes in the system characteristics and properties. Thus, it gave rise to the necessity of creating some new directions and branches in the automatic control theory: i.e., the theory of dynamic systems with delay, theory of discrete deterministic and stochastic processes, as well as the development of special methods of stability, controllability, and optimization [18], [87], [169] [199], [200]. However during further investigations of various technical, economic, and biological systems, the concept of impulsive control became known as an action causing the instantaneous changes in the system states. These changes occur much quicker than the proper dynamic processes and behave in such a manner that in the natural time scale these changes seem to be instantaneous. It was found natural to substitute these quick changes by jumps in mathematical models of such systems. This substitution made it possible to simplify the description of those systems. This simplification is obtained at the expense of reducing the dimension of the control actions while replacing the functional description by a parametrical one. Various examples of such systems were considered in flight dynamics [81], [82], [98], [100], [110], [117], [154], [155], [156], [173], [202], [215], [216], in optimizing the tactics of chemical and radio-therapy for cancer and other diseases [80], [166], in economic analysis [14], [46], [69], [70], in control of information processes [31], [36], [66], [107], [114], [115], and in queuing systems [108], [116]. The number of various areas of the impulsive control applications was also presented in monographs [12], [57], where besides those mentioned above the following examples are presented: power production control, stocks management.