# Find Lyapunov function.

#### dimvssometimes

##### New member
Hello. I have the interesting task in control theory. There is a theorem for systems like this:
$$\displaystyle \dot x = Ax + B \sin\theta \\ \dot \theta = Cx + D\sin\theta \\ x\in \mathbb{R}^n, \theta \in \mathbb{R}$$
A,B,C,D are matrices of corresponding sizes.
$$\displaystyle Let\ V(x,\theta)\ is\ scalar\ function\ such\ that: \\ \circ \ V(x, \theta + 2\pi) = V(x,\theta) \forall \ x\in \mathbb{R}^n, \theta \in \mathbb{R} \\ \circ \ V(x,\theta) \to \infty \ when\ |x| \to \infty \\ \circ \ For \ all \ solution \ (x(t),\theta(t)) \ of \ our \ system \ \dot V(x(t),\theta(t)) \le 0 \\ \circ \ If \ \dot V(x(t),\theta(t)) = 0, \dot (x(t),\theta(t)) \ \ is \ equilibrium\ position. \\ Then \ all \ solutions\ of\ our\ system\ go\ to\ set\ of\ equilibrium\ positions\ when\ t \to \infty.$$
I have the system which is below:
$$\displaystyle \dot x_1 = \sin \theta, \\ \dot x_2 = x_1, \\ \dot \theta = K(\tau_{z1}+\tau_{z_2})x_1 + Kx_2 + K\tau_{z1} \tau_{z_2} \sin \theta, \\ K>0, \ 0<\tau_{z1}<\tau_{z_2}<1.$$
The task is to find function $$\displaystyle V$$ which satisfies theorem for certain $$\displaystyle K,\tau_{z1},\tau_{z2}$$. I have tried to find V with form $$\displaystyle V = ax_1^2 + bx_2^2+c(1-\cos \theta)$$ but I did not succeed. Please, help me with this task.