Browsing by Author "Bivziuk, V. O."

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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.
Robust Stabilization of Non-Linear Non-Autonomous Control Systems With Periodic Linear Approximation
(Oxford Univ Press, 2021) Slyn'ko, V., I; Tunc, Cemil; Bivziuk, V. O.
The paper deals with the problem of stabilizing the equilibrium states of a family of non-linear non-autonomous systems. It is assumed that the nominal system is a linear controlled system with periodic coefficients. For the nominal controlled system, a new method for constructing a Lyapunov function in the quadratic form with a variable matrix is proposed. This matrix is defined as an approximate solution of the Lyapunov matrix differential equation in the form of a piecewise exponential function based on partial sums of a W. Magnus series. A stabilizing control in the form of a linear feedback with a piecewise constant periodic matrix is constructed. This control simultaneously stabilizes the considered family of systems. The estimates of the domain of attraction of an asymptotically stable equilibrium state of a closed-loop system that are common for all systems are obtained. A numerical example is given.