Tubes of discontinuous solutions of dynamical systems and their stability

The article is devoted to investigation of nonlinear dynamical systems which applies the generalized effect. The generalized effect (or impulses) is the result of the presence on the right part of the system a generalized derivative of the function of bounded variation. Such system contains incorrect operation of multiplication of discontinuous function on the generalized function from the point of view of the theory of distributions. This incorrectness is overcome by the approximation of the generalized functions in the right part of system by the sequence of smooth approximations of the generalized influences by analogy with sequential approach of the theory of the generalized functions. This sequence generates a sequence of smooth solutions. Then limit of a sequence of smooth solutions is considered. If such limit exists it is offered to be used as a solution. The solution is understood as partial pointwise limit of such sequence if a sequence of smooth solutions does not converge. Such partial limits constitute a tube of solutions. The sufficient conditions are received for stability of tubes of discontinuous solutions. © 2017 Author(s).Russian Science Foundation, RSFThe research was supported by Russian Science Foundation (RSF)(project No.16-11-10146)

On the stability of discontinuous solutions of bilinear systems with impulse action, constant and linear delays

This paper is devoted to the study of the stability properties of solutions bilinear system of differential equations with generalized effects in the system matrix, constant and linear delays in phase coordinates. Sufficient stability conditions are obtained. © 2019 Author(s).Russian Foundation for Basic Research, RFBR: 19-01-00371The research was supported by Russian Foundation for Basic Research, project no. 19-01-00371 and by Act 211 Government of the Russian Federation, contract 02.A03.21.0006

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

On pulse optimal control of linear systems with aftereffect

The paper deals with the problem of pulse optimal control of a linear dynamic system with aftereffect. As a functional, the degenerate quadratic functional of the most common kind is considered. The absence of control in the functional leads to the fact that the optimal control contains a pulse component. Sufficient conditions of existence of the pulse optimal control are obtained and the equations describing coefficients in the optimal control are worked out. Sufficient conditions that make it possible to integrate equations and to find the coefficients in an explicit form for the optimal control are established. A model example is considered. © 2013 Pleiades Publishing, Ltd

Singular linear-quadratic problem with a terminal condition

A singular linear-quadratic optimization problem with a terminal condition is considered. This problem in the space of absolutely continuous functions has no solutions. To ensure the existence of a solution, it is necessary to extend the set of admissible controls. In the case under consideration, we consider the generalized derivatives of functions of bounded variation as admissible controls. As a result, valid controls may contain impulse components. Under some assumptions, an optimal control is constructed containing impulse components at the initial and final instants of time. An illustrative example is provided. © 2021 Author(s).The research was supported by Russian Foundation for Basic Research, project no. 19-01-00371 and by Act 211 Government of the Russian Federation, contract 02.A03.21.0006

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

HYERS-ULAM-RASSIAS STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS WITH A GENERALIZED ACTIONS ON THE RIGHT-HAND SIDE

The paper considers the Hyers-Ulam-Rassias stability for systems of nonlinear differential equations with a generalized action on the right-hand side, for example, containing impulses - delta functions. The fact that the derivatives in the equation are considered distributions required a correction of the well-known Hyers-Ulam-Rassias definition of stability for such equations. Sufficient conditions are obtained that ensure the property under study.This work was supported by the Russian Science Foundation (project no. 22-21-00714)

On the stability of tubes of discontinuous solutions of bilinear systems with delay

The paper considers the stability property of tubes of discontinuous solutions of a bilinear system with a generalized action on the right-hand side and delay. A feature of the system under consideration is that a generalized (impulsive) effect is possible non-unique reaction of the system. As a result, the unique generalized action gives rise to a certain set of discontinuous solutions, which in the work will be called the tube of discontinuous solutions.The concept of stability of discontinuous solutions tubes is formalized. Two versions of sufficient conditions for asymptotic stability are obtained. In the first case, the stability of the system is ensured by the stability property of a homogeneous system without delay; in the second case, the stability property is ensured by the stability property of a homogeneous system with delay. These results generalized the similar results for systems without delay. © 2020 Irkutsk State University. All rights reserved

Impulse–Sliding Regimes in Systems with Delay

The paper is devoted to the formalization of a concept of impulse-sliding regimes generated by positional impulse controls for systems with delay. We define the notion of impulse-sliding trajectory as a limit of a sequence of Euler polygonal lines generated by a discrete approximation of the impulse position control. The equations describing the trajectory of impulse-sliding regime are received.The research was supported by Russian Science Foundation (RSF) (project No. 16-11-10146)