Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

• 成果类型: Article
• 署名作者: Chen, Can
• 署名单位: University of Michigan System; University of Michigan; Harvard University; Harvard University Medical Affiliates; Brigham & Women's Hospital; Harvard University; Harvard Medical School
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3209438
• 发表日期: 2023
• 页码: 4962-4969
• 关键词: Explicit solutions; homogeneous polynomial dy-namical systems (HPDS); orthogonal decomposition; STABILITY; ten-sor algebra; Z-eigenvalues; Z-eigenvectors

摘要

In this article, 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 that 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.