# Lyapunov Stability for a Nonlinear, Time-varying system

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?

• You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation. – Kwin van der Veen Dec 10 '18 at 9:43
• I think you might have mistaken a $-$ sign with $+$ – polfosol Feb 4 at 21:27