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How to construct lyapunov function?

Lyapunov equation

Lyapunov function

Square (algebra)

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Lyapunov functions can be constructed for polynomial dynamical systems using different methods. One approach is to use quantifier elimination to simplify the problem of finding suitable Lyapunov functions for polynomial systems . Another method is to use Constrained Least Square Optimization to determine the coefficients of the Lyapunov function, ensuring its positive definiteness and negative semi-definiteness of its derivative . Additionally, a novel vector field decomposition based approach can be used, where the potential function of a decomposed vector field can serve as a Lyapunov function candidate . This approach is applicable to systems with a decomposition into two mutually orthogonal vector fields, one of which is curl-free and the other is divergence-free. The existence of this decomposition can be determined by solving specific equations .

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Journal Article•DOI Construction of Lyapunov Functions Using Vector Field Decomposition Yuanyuan Liu 13 Jul 2022-arXiv.org	The paper proposes a novel approach for constructing Lyapunov functions based on vector field decomposition. If the vector field can be decomposed into two mutually orthogonal fields, one curl-free and the other divergence-free, the potential function of the curl-free field can serve as a Lyapunov function candidate.
Proceedings Article•DOI Lyapunov Function Construction using Constrained Least Square Optimization M. Wasim, Desineni Subbaram Naidu - Show less +1 more 17 Oct 2022	The paper proposes a method to construct a Lyapunov function for polynomial dynamical systems using Constrained Least Square Optimization. The method involves selecting a polynomial Lyapunov function with unknown coefficients and solving a least-square problem to determine the coefficients that make the function positive definite and its derivative negative semi-definite.
Open access•Journal Article•DOI On the Systematic Construction of Lyapunov Functions for Polynomial Systems D Gerbet, Klaus Röbenack - Show less +1 more 01 May 2023-Pamm	The paper proposes using quantifier elimination and a parametric ansatz to construct Lyapunov functions for polynomial systems. It also suggests simplifying the problem by deriving easier to evaluate necessary conditions.
Proceedings Article•DOI Lyapunov Function Construction using Constrained Least Square Optimization 17 Oct 2022	The paper proposes a method to construct a Lyapunov function for polynomial dynamical systems using Constrained Least Square Optimization.

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