Impulse position control for differential inclusions

For a nonlinear control system, presented in the form of a differential inclusion with impulse control, the concept of the impulse-sliding regime generated by the positional impulse control is defined. The basis of formalization is a discrete scheme. It is shown that the impulse-sliding regime satisfies some differential inclusion. Illustrative examples are given. © 2018 Author(s).The research was supported by Russian Foundation for Basic Research, project no. 16-01-00505

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

Method of Limiting Differential Inclusions and Asymptotic Behavior of Systems with Relay Controls

In this paper, problems of asymptotic behavior of non-autonomous controlled systems with a matrix of derivatives and the feedbacks of relay type are considered. The research is based on the method of limiting equations in combination with the direct method of Lyapunov functions with semidefinite derivatives. The method of the limiting equations has arisen in works G.R. Sell (1967) and Z. Artstein (1977, 1978) on topological dynamics of nonautonomous systems. Now this method is advanced for a wide class of systems, including the systems with delay. Nevertheless the method of the limiting equations till now has not received development with reference to nonautonomous differential inclusions and discontinuous systems for which it has fragmentary character. The main results are bound up with development of this method for discontinuous systems represented in the form of differential inclusions. In this case, specific methods of multivalued analysis and development of new methods for constructing limiting differential inclusions were required. The structure of the systems under scrutiny makes it possible, in particular, to study mechanical systems controlled on the decomposition principle of E.S. Pyatnitsky, and systems with dry friction submitted by equations Lagrange of 2-nd kind

Positional impulse and discontinuous controls for differential inclusion

Nonlinear control systems presented in the form of differential inclusions with impulse or dis- continuous positional controls are investigated. The formalization of the impulse-sliding regime is carried out. In terms of the jump function of the impulse control, the differential inclusion is written for the ideal impulse- sliding regime. The method of equivalent control for differential inclusion with discontinuous positional controls is used to solve the question of the existence of a discontinuous system for which the ideal impulse-sliding regime is the usual sliding regime. The possibility of the combined use of the impulse-sliding and sliding regimes as control actions in those situations when there are not enough control resources for the latter is discussed. © 2020, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was supported by Russian Foundation for Basic Research (project No. 19-01-00371

Limiting Differential Inclusions and the Method of Lyapunov’s Functions

In article the method of research of asymptotic behaviour for solutions of the nonautonomous systems submitted in the form of differential inclusions develops. The received results carry the form of generalizations of the LaSalle’s principle of invariance . Principle of invariance usually call as LaSalle’s theorem for the autonomous differential equations in which (in the frame of Lyapunov’s direct method) it is supposed, that derivative of Lyapunov’s function is nonpositivity. The conclusion which this implies, will be, that the right limiting sets of solutions belong to the greatest invariant subset from set of zero of derivative function of Lyapunov. Before Lyapunov’s functions with constant signs were used in Barbashin – Krasovsky’s known theorem about asymptotic stability of positions of balance of autonomous systems. This theorem (together with LaSalle’s theorem) also sometimes characterize, as a principle of invariancy. For the nonautonomous equations on this way there are the difficulties connected to absence of properties such as invariancy of the right limiting sets of solutions, and also with the description of set of zero of a derivative of Lyapunov’s functions. Attempts of overcoming of these difficulties have led to concept of the limiting differential equations. Method of the limiting equations in a combination to Lyapunov’s direct method allows to investigate effectively asymptotic behaviour of solution of nonautonomous systems. These researches go back to works of G.R. Selll and Z. Artstein on topological dynamics of the nonautonomous differential equations. Distribution of a method of the limiting equations on wider classes of systems brings an attention to the question about structure and methods of construction of the limiting equations. We this question is solved with reference to differential inclusion