On the Application of the Wa\.zewski Method to the Problem of Global Stabilization

TL;DR

This paper explores how the Ważewski topological method can be applied to feedback control and dynamical systems to prove the impossibility of global stabilization and identify conditions for trajectories that remain bounded.

Contribution

It introduces a novel application of the Ważewski method to control systems, providing criteria for non-asymptotic trajectories and demonstrating with real-world examples.

Findings

Proves the impossibility of global stabilization in certain systems.

Provides sufficient conditions for trajectories to remain within a subset.

Illustrates results with Furuta pendulum and inverted pendulum examples.

Abstract

We consider a possible application of the Wa\.zewski topological method to feedback control systems and to more general dynamical systems. We show how this method can be used to prove the impossibility of global stabilization in such problems. Moreover, we give sufficient conditions for the existence of a solution such that its trajectory never leaves a subset of the extended phase space of the system and does not tend asymptotically to a given equilibrium. We illustrate our result with various real-life systems including the Furuta pendulum and the wheeled inverted pendulum.