Open AccessFunctional Method of Localization and LaSalle Invariance Principle
Author(s) -
А. Н. Канатников,
А. П. Крищенко
Publication year - 2021
Publication title -
matematika i matematičeskoe modelirovanie
Language(s) - English
Resource type - Journals
ISSN - 2412-5911
DOI - 10.24108/mathm.0121.0000256
Subject(s) - invariance principle , attractor , invariant (physics) , limit set , dynamical systems theory , mathematics , phase space , compact space , dynamical system (definition) , stability (learning theory) , set (abstract data type) , lyapunov function , limit (mathematics) , mathematical analysis , computer science , nonlinear system , physics , philosophy , linguistics , quantum mechanics , machine learning , mathematical physics , thermodynamics , programming language
A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances. The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space. Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions. Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.