Non-Anticipating Procedures of Control and Iteration Methods for Constructing Them

Abstract

In the previous chapters different extension constructions for natural problems of attainability are considered. These problems are connected with extremal problems; this connection is discussed in detail in [32, 35] and many other investigations. In fact, the constructions of Chapters 3 and 4 are concerned with the employment of FAM for extensions and can be regarded as constructions of the asymptotic optimization. But very often the natural optimization is complicated by noise. These difficulties arise in many dynamic problems. So it is possible to consider the problems of control under the action of a noise. The given direction in control theory, being the important part of this theory, is called differential games theory. The structure of differential games is defined by the well known theorem about an alternative of Krasovskii and Subbotin. The above mentioned alternative was established in the class of very non-regular positional strategies. In addition, two players participate in the game. One player commands a useful control. The other player forms a hindrance. For constructing their own controls the players use feedback. But in the problems (of guidance and evasion) non-linear and discontinuous laws of feedback are required. The realization of such a feedback in the form of positional strategies requires a distinctive extension relative to the formations of trajectories. Namely, the so called constructive motions of Krasovskii are required here.