Browsing by Author "Slyn'ko, V. I."

Now showing 1 - 4 of 4

Application of Commutator Calculus To the Study of Linear Impulsive Systems
(Elsevier Science Bv, 2019) Slyn'ko, V. I.; Tunc, Osman; Bivziuk, V. O.
In this paper, the formulas of commutator calculus are applied to the investigation of the stability of linear impulsive differential equations. It is assumed that the moments of impulse action satisfy the average dwell-time (ADT) condition. Sufficient conditions for the asymptotic stability of linear impulsive differential equations in a Banach space are obtained. In the Hilbert space, the stability of the original linear differential equation is reduced to the investigation of a linear differential equation with equidistant moments of impulse action and perturbed discrete dynamics. This reduction simplifies the application of Lyapunov's direct method and the construction of Lyapunov functions. We give examples in the spaces R-2 and X = C[0, l] to illustrate the effectiveness of results obtained. Finally, a sufficient generality of the obtained results on the dynamic properties of linear operators of the linear impulsive differential equation is established. (C) 2018 Elsevier B.V. All rights reserved.
Global Asymptotic Stability of Nonlinear Periodic Impulsive Equations
(Univ Miskolc inst Math, 2018) Slyn'ko, V. I.; Tunc, Cemil
Pseudo-linear impulsive differential equations in a Banach space are considered. It is assumed that the conditions of a small change in the operator coefficients of the equation are satisfied. Using the method of "frozen" coefficients and the methods of commutator calculus, the problem of global asymptotic stability of a pseudo-linear impulsive differential equation is reduced to the problem of estimating the evolution operator for linear impulsive differential equation with constant operator coefficients. The obtained results are applied for stability study of a nonlinear system of ordinary impulsive differential equations. Lyapunov's direct method is used for estimating the fundamental matrix of the corresponding system of impulsive differential equations with constant coefficients. The stability conditions are formulated in terms of the solvability of certain linear matrix inequalities.
Instability of Set Differential Equations
(Academic Press inc Elsevier Science, 2018) Slyn'ko, V. I.; Tunc, Cemil
This paper is devoted to the instability of Set Differential Equations (SDEs). Using the geometric inequalities of Brunn-Minkowski and A.D. Aleksandrov, we propose new methods for constructing Lyapunov functions. In combination with the known methods of stability theory, the Lyapunov's direct method, the comparison method and the vector-function method, we establish conditions for the collapse of the solutions of the SDEs. Estimates of the collapse time of solutions are also obtained. Examples of SDEs in spaces of dimension 2 and 3 illustrating general theorems are given. (C) 2018 Published by Elsevier Inc.
Sufficient Conditions for Stability of Periodic Linear Impulsive Delay Systems
(Maik Nauka/interperiodica/springer, 2018) Slyn'ko, V. I.; Tunc, Cemil
For a linear periodic system with impulsive action and delay, new approaches to the study of stability were proposed on the basis of the methods of spectral theory of linear operators, direct Lyapunov method, and N.G. Chetaev method for construction of the Lyapunov functions for the periodic linear systems, as well as the perturbation method for construction of the Lyapunov functions. These methods underlie the sufficient conditions for asymptotic stability of the linear periodic systems with impulsive action and delay. We gave some illustrative examples of studying stability of such systems under different assumptions about the dynamic properties of the continuous and discrete components of the impulsive system.