[Paper Review] The Lyapunov Concept of Stability from the Standpoint of Poincare Approach: General Procedure of Utilization of Lyapunov Functions for Non-Linear Non-Autonomous Parametric Differential Inclusions

Myroslav Sparavalo|arXiv (Cornell University)|Mar 23, 2014

Control and Stability of Dynamical Systems2 references3 citations

TL;DR

This paper introduces a general procedure for constructing Lyapunov functions for nonlinear non-autonomous parametric differential inclusions using a Poincaré-inspired three-point strategy and geometric-topological analysis via hierarchical fiber bundles. It establishes canonical forms through a special transformation, enabling standard Lyapunov functions to be applied, and derives conditions for local and global asymptotic stability, particularly through the classificational stability of typical fibers in a meta-bundle structure.

ABSTRACT

The objective of the research is to develop a general method of constructing Lyapunov functions for non-linear non-autonomous differential inclusions described by ordinary differential equations with parameters. The goal has been attained through the following ideas and tools. First, three-point Poincare strategy of the investigation of differential equations and manifolds has been used. Second, the geometric-topological structure of the non-linear non-autonomous parametric differential inclusions has been presented and analyzed in the framework of hierarchical fiber bundles. Third, a special canonizing transformation of the differential inclusions that allows to present them in special canonical form, for which certain standard forms of Lyapunov functions exist, has been found. The conditions establishing the relation between the local asymptotical stability of two corresponding particular integral curves of a given differential inclusion in its initial and canonical forms are ascertained. The global asymptotical stability of the entire free dynamical systems as some restrictions of a given parametric differential inclusion and the whole latter one per se has been investigated in terms of the classificational stability of the typical fiber of the meta-bundle. There have discussed the prospects of development and modifications of the Lyapunov second method in the light of the discovery of the new features of Lyapunov functions.

Motivation & Objective

• To develop a general method for constructing Lyapunov functions for nonlinear non-autonomous parametric differential inclusions.
• To apply the three-point Poincaré strategy to analyze the stability of differential inclusions and their integral curves.
• To analyze the geometric-topological structure of such inclusions using hierarchical fiber bundles and meta-bundles.
• To identify conditions under which the local asymptotic stability is preserved between the original and canonical forms of the differential inclusion.
• To investigate global asymptotic stability of free dynamical systems and the full parametric inclusion through the classificational stability of typical fibers.

Proposed method

• Employing the three-point Poincaré strategy to study the behavior of solutions and manifolds in non-autonomous differential inclusions.
• Representing the geometric-topological structure of the differential inclusions within the framework of hierarchical fiber bundles.
• Deriving a special canonizing transformation that converts the original differential inclusion into a canonical form where standard Lyapunov functions are applicable.
• Establishing necessary and sufficient conditions for the preservation of local asymptotic stability between corresponding integral curves in the original and canonical forms.
• Analyzing global asymptotic stability by examining the classificational stability of the typical fiber within the meta-bundle structure.
• Using tools from differential topology, fiber bundle theory, and stability theory to unify the treatment of parametric and non-autonomous systems.

Experimental results

Research questions

• RQ1 How can the Poincaré three-point strategy be adapted to analyze stability in nonlinear non-autonomous parametric differential inclusions?
• RQ2 What canonical form can be achieved via a special transformation to enable the use of standard Lyapunov functions?
• RQ3 What conditions ensure that local asymptotic stability is preserved between the original and canonical forms of the differential inclusion?
• RQ4 How can global asymptotic stability of the entire free dynamical system be characterized in terms of fiber bundle structure?
• RQ5 What role does the classificational stability of the typical fiber in the meta-bundle play in determining the stability of the full parametric system?

Key findings

• A general procedure for constructing Lyapunov functions is established for nonlinear non-autonomous parametric differential inclusions through a canonizing transformation.
• Conditions are derived that guarantee the preservation of local asymptotic stability between corresponding integral curves in the original and canonical forms of the inclusion.
• The geometric-topological structure of the system is fully characterized using hierarchical fiber bundles, enabling a systematic stability analysis.
• Global asymptotic stability of the entire free dynamical system is linked to the classificational stability of the typical fiber in the meta-bundle.
• The study reveals new structural features of Lyapunov functions, suggesting extensions and modifications to the classical Lyapunov second method.
• The framework provides a unified approach to stability analysis that integrates differential equations, topology, and control theory.