Nov 26, 2016 · I'm trying to understand the difference between general Lyapunov stability and uniform stability. I understand, that Lyapunov s. is the situation, when for every ϵ> 0 ϵ> 0, there exists δ> 0 δ> 0, such that when the solution starts within a distance δ δ from the equilibrium point, it remains within a distance ϵ ϵ from it forever.

May 30, 2020 · Using definition of Lyapunov's stability, check if following solution is stable:

The origin being Lyapunov stable for x˙ = Ax x ˙ = A x does not necessarily imply that it remains Lyapunov stable when perturbed by a term O(∥x∥2) O (‖ x ‖ 2) as in x˙ = v(x) x ˙ = v (x). Your example demonstrates that it is possible for the stability to change when nonlinear terms are added to the system.

Aug 16, 2023 · where x ∈ Rn and A ∈Rn×n In the textbook LMIs in Control Systems: Analysis, Design and Applications by Duan and Yu, the following condition is given for the stability of (1) using Lyapunov stability

Feb 20, 2018 · I'm having some trouble grasping the difference between attractivity and Lyapunov stability. Is it always the case that an equilibrium of a function which is attractive is also Lyapunov stable, an...

May 3, 2020 · For a linear system it does provide necessary and sufficient conditions for stability, for a general non-linear system it's not the case because failure of a chosen Lyapunov function to satisfy all the conditions for stability does not mean that the corresponding equilibrium point is not stable. It (may) just indicate that the chosen Lyapunov function is not good enough. For …

Jul 19, 2020 · I grapple here with the same question. So maybe this helps. I think your definition excludes the requirement that δ> 0 δ> 0. If x(0) =x∗ x (0) = x ∗ then your definition will be Lyapunov stable whereas the Wikipedia definition would not.

May 5, 2015 · For asymptotic stability, we don't need the exact values of eigenvalues, we only need that their real part are strictly negative. The existence of P =P∗> 0 P = P ∗> 0 satisfying the Lyapunov equation is a sufficient criterion. We have explicit formulas for solutions of …

Jan 22, 2017 · This problem can be handled with an optimization procedure, having in mind that generally is a non convex problem. The result depends on the test Lyapunov function used so we will generalize to a quadratic Lyapunov function

The Lyapunov theorem as I know it requires the Lyapunov function to have a strict local minimum at the equilibrium point. Yours doesn't, hence the theorem is not applicable. And your reasoning showing that is certainly spot on.