I have this system and i want study stability: $$ \left\{ \begin{array}{c} \dot x_1 = x_3 \\ \dot x_2=x_4\\ \dot x_3 =\frac{1}{I}[u-bx_3-k(x_1-x_2)]\\ \dot x_4 =\frac{1}{mL^2}[k(x_1-x_2)-mgL\sin(x_2)] \end{array} \right. $$
I can't use phase portrait or Poincaré map because this have 4-dimension. I try to find Lyapunov candidate but for now nothing. Can help me?
Since $u$ is not given and for simplicity's sake, we assume that $u = 0$, $I = 1$, $m = 1$, $L = 1$, $b > 0$, $k > 0$, $g > 0$, and $0 < |x_{2}| < \pi$, then with the following Lyapunov function candidate
$$ V\left(\mathbf{x}\right) = \begin{bmatrix} x_{1} & x_{2} & x_{3} & x_{4} \\ \end{bmatrix} \begin{bmatrix} p_{11} & p_{12} & p_{13} & p_{14} \\ p_{12} & p_{22} & p_{23} & p_{24} \\ p_{13} & p_{23} & p_{33} & p_{34} \\ p_{14} & p_{24} & p_{34} & p_{44} \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{bmatrix} $$
where
$$ p_{11} = \frac{b^2 g k + b^2 g + 2 b^2 k + g^2 k + 2 g k^2 + 2 g k}{2 b g k} $$
$$ p_{12} = -\frac{b^2 g + b^2 k + b^2 + g^2 + g k + g + 2 k^2}{2 b k} $$
$$ p_{13} = \frac{g + 2 k + g k}{2 g k} $$
$$ p_{14} = \frac{g + 2}{2 g} $$
$$ p_{22} = \frac{b^2 g^2 + 2 b^2 g k + b^2 g + b^2 k^2 + b^2 k + g^3 + g^2 k + g^2 + 3 g k^2 + 2 k^3 + 2 k^2}{2 b k^2} $$
$$ p_{23} = -\frac{g + k + 1}{2 k} $$
$$ p_{24} = -\frac{1}{2} $$
$$ p_{33} = \frac{g + 2 k + 2 g k}{2 b g k} $$
$$ p_{34} = -\frac{g^2 + g - 2 k}{2 b g k} $$
$$ p_{44} = \frac{b^2 g^2 + b^2 g k + b^2 g + g^3 + g^2 + 2 g k^2 - g k + 2 k^2}{2 b g k^2} $$
I think it is possible to show to $V\left(\mathbf{x}\right)$ is positive definite and $\dot{V}\left(\mathbf{x}\right)$ is negative definite over $0 < |x_{2}| < \pi$.