Overview of V.A. Plotnikov’s research on averaging of differential inclusions

Abstract

In this review we will first look in detail at V.A. Plotnikov's results on the substantiation of full and partial schemes of averaging for differential inclusions in the standard form on final and infinite interval. Then we will consider the algorithms where there is no average, but there is a possibility to find its estimation from below and from above. Such approach is also used when the detection of an average is approximate. This situation is especially typical at consideration of differential inclusions with fast and slow variables. In the last part we will give the results concerning the substantiation of the full and partial averaging method for impulsive differential inclusions on final and infinite intervals.

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

“…The justification of the averaging method for such systems in terms of Rsolutions was given by Plotnikov & Plotnikova [98]. A detailed account of the works on this subject can be found in the book by Perestyuk et al [77] and in Klymchuk et al [61]. Skripnik [109] extended the averaging method to impulsive differential inclusions with Hukuhara derivative.…”

“…In fact, averaging method for differential inclusions suffered a major development with the works of Plotnikov and his school. For an account of this results, we refer the reader to the books Plotnikov [99] and Plotnikov et al [94], and a recent survey of Plotnikov's research, Klymchuk et al [61].…”

“…In this survey we give a detailed account of the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions. We emphasize the results not explored in the survey by Klymchuk et al [61]. More precisely, the paper is organized as follows.…”

Tight Bounds for Averaging Multi-Frequency Differential Inclusions, Applied to Control Systems

“…For a reference on the averaging method of ordinary differential equations, the reader is referred to the book of Sanders, Verhulst and Morduck [10] and to works of Artstein [1] and Bright [5,6] for a modern treaty and improved estimates on the error, the line of which we follow in this paper. For a reference to results in differential inclusions refer to the review papers Klymchuk, Plotnikov and Skripnik [8] and to [7].…”

Weak Exponential Stability for Time-Periodic Differential Inclusions via First Approximation Averaging

“…The averaging of the first approximation allows us to get easy verifiable sufficient exponential stability conditions and to make the proof more natural and direct. Some other results concerning weak stability for differential inclusions via the averaging method can be found in the survey [10].…”