Find Lyapunov function.

dimvssometimes

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Feb 27, 2020
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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.
 
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