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?

  • $\begingroup$ 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

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