Related papers: Construction of Lyapunov Functions Using Vector Fi…
The Helmholtz-Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function.…
This article presents a novel numerically tractable technique for synthesizing Lyapunov functions for equilibria of nonlinear vector fields. In broad strokes, corresponding to an isolated equilibrium point of a given vector field, a…
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of…
We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector…
In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the…
This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential…
Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been…
Lyapunov functions play a fundamental role in analyzing the stability and convergence properties of optimization methods. In this paper, we propose a novel and straightforward approach for constructing Lyapunov functions for first-order…
We consider constructing Lyapunov functions for systems that are both monotone and contractive with respect to a weighted one norm or infinity norm. This class of systems admits separable Lyapunov functions that are either the sum or the…
Control Lyapunov functions (CLFs) play a vital role in modern control applications, but finding them remains a problem. Recently, the control Lyapunov-value function (CLVF) and robust CLVF have been proposed as solutions for nonlinear…
A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as…
The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing (cooperative) and decreasing (competitive) components, and this decomposition allows for, e.g., efficient computation of reachable sets…
This paper studies the use of vector Lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. Recent results on vector Lyapunov functions are utilized. The main result of the paper shows that the…
This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means.…
Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we…
This paper considers a wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable vector of state. The problem is to find a special function (Lyapunov function), which in the framework of the second…
This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis and the global, operator theoretic, Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov…
We study convergence of nonlinear systems in the presence of an `almost Lyapunov' function which, unlike the classical Lyapunov function, is allowed to be nondecreasing---and even increasing---on a nontrivial subset of the phase space.…
This paper presents a new interpretation of the Lyapunov and Riccati equations from the perspective of positive system theory. We show it is possible to construct positive systems related to these equations, and then certain conclusions --…
Monotone systems preserve a partial ordering of states along system trajectories and are often amenable to separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a scalar argument. In this paper,…