Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

TL;DR

This paper develops a tensor algebra-based framework for explicitly solving and analyzing the stability of homogeneous polynomial dynamical systems, especially those represented by orthogonally decomposable tensors, with applications to control and approximation.

Contribution

It introduces a method to explicitly solve odeco homogeneous polynomial systems and analyze their stability using tensor Z-eigenvalues, extending linear system concepts.

Findings

Explicit solutions for odeco HPDS using tensor Z-eigenvalues

Necessary and sufficient stability conditions derived from eigenvalues

Framework for approximating general HPDS by odeco systems

Abstract

In this paper, we provide a system-theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonally decomposable (odeco) tensor, we can construct its explicit solution by exploiting tensor Z-eigenvalues and Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of the explicit solution, we are able to discuss the stability properties of an odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic tensor can be used to establish necessary and sufficient stability conditions, similar to these from linear systems theory. In addition, we are able to obtain the complete solution to an odeco HPDS with constant control. Finally, we establish results which enable one to determine if a general HPDS can be transformed to or approximated by an odeco HPDS, where the previous results can be applied. We demonstrate our framework with simulated and real-world examples.