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The nonlinear system: $\dot{x}_1 = \frac{5}{2}-2x_2-\frac{3}{2}x_1+x_2x_1$ and $\dot{x}_2 = x_1-1$.

How do I find the Lyapunov function of this system? Or how to determine the existence of such a function? I know there's an equilibrium point at $(1, 1)$ and it's stable. Does that indicate the existence of a Lyapunov function? Any answers or further discussion would be great.

Desperate Fluffy
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  • Do you just want to find a Lyapunov function such that the equilibrium point $(1,1)$ is asymptotically stable? – KittyL Jun 18 '15 at 18:04

1 Answers1

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We can find a Liapunov function around the equilibrium point $(1,1)$ so that it can be shown to be asymptotically stable.

Let $V(x_1,x_2)=a(x_1-1)^2+b(x_2-1)^2$. Then

$$\dot{V}=2a(x_1-1)\dot{x_1}+2b(x_2-1)\dot{x_2}\\ =5ax_1-2bx_1-4ax_1x_2-3ax_1^2+2bx_1x_2+2ax_1^2x_2-5a+2b+4ax_2+3ax_1-2bx_2-2ax_1x_2$$

Let $b=3a$ to eliminate the $x_1x_2$ term. Then let $a=1$ since each term contains an $a$ and we want the square term to be negative.

$$\dot{V}=(-3+2x_2)x_1^2+2x_1-6x_2+1$$

Now if you choose $\frac{3}{4}<x_2<\frac{5}{4}$, $\frac{3}{4}<x_1<\frac{5}{4}$, it would be negative.

KittyL
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