3
$\begingroup$

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.

Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[x_1(t)+g(t)]$$ $$\dot{x_2}(t)= x_1(t)[x_1(t)+g(t)]$$ And we want to investigate the stability of the solution $x(t)=0$.

If we use a simple Lyapunov function $$V(x) = 0.5x_{1}^{2} + 0.5x_{2}^{2}$$

I can find $\dot{V}(x,t)$, but I am unsure of where to go from here. How do I prove some kind of stability/instability. Do I need Barbalat's Lemma?

$\endgroup$
  • $\begingroup$ You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation. $\endgroup$ – Kwin van der Veen Dec 10 '18 at 9:43

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.