# How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

\left.\begin{aligned} \dot{x_1} &= x_2 \\ \dot{x_2} &= -\frac{x_1}{1 + x_2^2} \label{eq:q2} \end{aligned}\qquad\right\}

• Are you familiar with Lyapunov functions? And what have you tried yourself? – Kwin van der Veen Nov 12 '18 at 16:58
• I have tried general quadratic functions, such as (x1^2 + x2^2), but a term xy remains and I can't conclude properly. – Abhinav Sinha Nov 12 '18 at 17:25

## 1 Answer

You could try to get the Lyapunov function in the form $$V=\frac12x_1^2+g(x_2)$$ so that $$\dot V=x_1x_2+g'(x_2)\frac{-x_1}{1+x_2^2}$$ so that one would get a usable result with $$g'(x_2)=x_2(1+x_2^2)$$, integrating to,for example, $$g(x_2)=\frac14(1+x_2^2)^2$$.

• Thank you so much. I was taking something similar to g(x2) but it was constant and I was stuck. Thanks for your help :) – Abhinav Sinha Nov 12 '18 at 18:55
• @SampleTime : Then consider $V-V(0)$ to get the normalized value. I prefer the completed square for its simplicity. – LutzL Nov 12 '18 at 21:18
• @SampleTime : Could you elaborate? How to get a negative value from $V-V(0)=\frac12x_1^2+\frac12x_2^2+\frac14x_2^4$? – LutzL Nov 12 '18 at 21:27
• @LutzL You are correct, nice! I made an error when writing down the equation, sorry for the confusion. – SampleTime Nov 12 '18 at 21:40