Work

A Converse Sum of Squares Lyapunov Result with a Degree Bound

Year: 2012

Type: preprint

Abstract: Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.1201.2619

Authors: Peet, Matthew M., Papachristodoulou, Antonis

Institutions: Illinois Institute of Technology, University of Oxford

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Engineering and Physical Sciences Research Council, Division of Civil, Mechanical and Manufacturing Innovation

Awards: Autonomous Behaviour and Learning in an Uncertain World, Using Control Theory to Design Sustainable Policies for Greenhouse Gas Emissions in the Presence of Model Uncertainty, Control Engineering Inspired Design Tools for Synthetic Biology, G8511057088

OpenAlex ID: W2951228608

Cites: 26

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Related to: 10