Construction of a stabilizing control and solution to a problem about the center and focus for differential systems with a polynomial part on the right side

TL;DR

This paper establishes conditions for the stability of polynomial differential systems and explores how to construct polynomial controls to stabilize such systems, focusing on the origin as a center or focus.

Contribution

It provides necessary and sufficient conditions for stability domains and introduces methods for designing polynomial stabilizing controls for these systems.

Findings

Conditions for asymptotic stability are formulated.

A method for constructing polynomial stabilizing controls is developed.

The analysis distinguishes between center and focus cases at the origin.

Abstract

Stationary differential systems with polynomial right sides are considered. Necessary and sufficient conditions are formulated when a given domain is a domain of asymptotic stability and the origin of coordinates is either focus or center. The problem of construction of a stabilizing control in the form of polynomial is studied.