TY - JOUR

T1 - О диагональной устойчивости некоторых классов сложных систем с запаздыванием

AU - Aleksandrov, A. Yu

AU - Vorob'Eva, A. A.

AU - Kolpak, E. P.

N1 - Publisher Copyright: © 2018 Saint Petersburg State University. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018

Y1 - 2018

N2 - The paper deals with the problem of diagonal stability of nonlinear difference-differential systems. Certain classes of complex systems with delay and nonlinearities of a sector type are studied. It is assumed that these systems describe the interaction of two-dimensional blockswith a delay in connections between the blocks. Two kinds of structure of connections are investigated. For every kind, necessary and sufficient conditions for the existence of diagonal Lyapunov-Krasovskii functionals are found. The existence of such functionals guarantees the asymptotic stability of the zero solutions of considered systems for any nonnegative delay and any admissible nonlinearities. These conditions are formulated in terms of the Hurwitz property of specially constructed Metzler matrices. The proposed approaches are used for the stability analysis ofsome models of population dynamics. Generalized Lotka-Volterra models composed of several interacting pairs of predator-prey type are investigated. With the aid of the Lyapunov direct method and diagonal Lyapunov-Krasovskii functionals, conditions are derived under which equilibrium positions of the considered models are globally asymptotically stable in the positive orthant of the state space for any nonnegative delay. An illustrative example and results of the numerical simulation are presented to demonstrate the effectiveness of the developed approaches.

AB - The paper deals with the problem of diagonal stability of nonlinear difference-differential systems. Certain classes of complex systems with delay and nonlinearities of a sector type are studied. It is assumed that these systems describe the interaction of two-dimensional blockswith a delay in connections between the blocks. Two kinds of structure of connections are investigated. For every kind, necessary and sufficient conditions for the existence of diagonal Lyapunov-Krasovskii functionals are found. The existence of such functionals guarantees the asymptotic stability of the zero solutions of considered systems for any nonnegative delay and any admissible nonlinearities. These conditions are formulated in terms of the Hurwitz property of specially constructed Metzler matrices. The proposed approaches are used for the stability analysis ofsome models of population dynamics. Generalized Lotka-Volterra models composed of several interacting pairs of predator-prey type are investigated. With the aid of the Lyapunov direct method and diagonal Lyapunov-Krasovskii functionals, conditions are derived under which equilibrium positions of the considered models are globally asymptotically stable in the positive orthant of the state space for any nonnegative delay. An illustrative example and results of the numerical simulation are presented to demonstrate the effectiveness of the developed approaches.

KW - Complex system

KW - Delay

KW - Diagonal stability

KW - Lyapunov-Krasovskii functional

KW - Population dynamics

UR - http://www.scopus.com/inward/record.url?scp=85090555682&partnerID=8YFLogxK

U2 - 10.21638/11701/SPBU10.2018.201

DO - 10.21638/11701/SPBU10.2018.201

M3 - статья

AN - SCOPUS:85090555682

VL - 15

SP - 72

EP - 88

JO - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1811-9905

IS - 4

ER -