Synopsis
This monograph presents an integral method for analysing the dynamic behaviour of nonlinear, non-stationary, and controlled systems. The method is based on the use of nonlinear integral inequalities to obtain new estimates for the norms of solutions and Lyapunov functions for the corresponding systems of differential equations of disturbed motion. The book consists of seven chapters. The first chapter establishes new bounds for solutions to ordinary and infinite systems of differential equations using nonlinear integral inequalities in pseudo-linear form. The second chapter studies the equations of disturbed motion based on new estimates of Lyapunov functions. Here, conditions are established for various types of motion boundedness, including the stability of coupled systems under initial and subsequent disturbances. The third chapter is devoted to polynomial systems, where variations of Lyapunov functions are used to derive conditions for stability and stabilisation of motion, including the analysis of the zero solution of systems with aftereffects. Chapter 4 applies nonlinear integral inequalities to nonlinear systems with interval initial conditions, and studies the stabilisation of systems with multiple controls. Chapter 5 focuses on quasilinear systems with fractional derivatives, establishing conditions for boundedness and Lagrange stability. Chapter 6 introduces an integral method for time-scale dynamic equations with fractional derivatives, offering new tools for stability and boundedness analysis. The final chapter studies equilibrium stability in a model of confrontation between two countries and alliances, using Lyapunov functions and integral inequalities to determine conditions for stable equilibrium and changes in weapon levels.
About the Author
Anatoliy A Martynyuk is a renowned specialist in the field of stability theories and nonlinear mechanics. In 1967, he defended his PhD thesis, and in 1973, his doctorate in physical and mathematical sciences at the Institute of Mathematics of the National Academy of Sciences of Ukraine. In 1977, he created the Department of Process Stability at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine and has been its head ever since. In 1992, he initiated the creation of the international series of scientific monographs Stability and Control: Theory, Methods and Applications (Great Britain). The series has published 22 volumes, which have received worldwide recognition. In 2001, Dr Martynyuk founded the international academic journal Nonlinear Dynamics and Systems Theory (with its online version at www.e-ndst.kiev.ua) and is its editor-in-chief. In 2006, he founded a new international series of scientific monographs, textbooks, and lecture courses entitled Stability, Oscillations and Optimization of Systems, published by Cambridge Scientific Publishers (UK), and is its editor-in-chief. To date, 11 volumes of the series have been published. He is the author (or co-author) of several monographs and books in English, Chinese, and Russian, devoted to the problems of stability and control of nonlinear dynamic systems. In 2008, Dr Martynyuk was awarded the State Prize of Ukraine in the field of Science and Technology, and in 2009, he was elected as a full member of the National Academy of Sciences of Ukraine. According to the 2024 ScholarGPS Top Scholar distinction, he is among the top 0.5% of scientists globally in nonlinear systems.
"About this title" may belong to another edition of this title.
